[ { "id": "Geometry_0000", "subject": "Geometry", "topic": "Shapes", "subtopic": "Perimeter", "level": "2", "keywords": [ "perimeter", "rectangle" ], "problem_v1": "A rectangle\u2019s perimeter is ${86\\ {\\rm cm}}$. Its base is ${28\\ {\\rm cm}}$.\nIts height is [ANS]cm. (Use cm for centimeters.)", "answer_v1": [ "15" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rectangle\u2019s perimeter is ${78\\ {\\rm cm}}$. Its base is ${20\\ {\\rm cm}}$.\nIts height is [ANS]cm. (Use cm for centimeters.)", "answer_v2": [ "19" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rectangle\u2019s perimeter is ${78\\ {\\rm cm}}$. Its base is ${23\\ {\\rm cm}}$.\nIts height is [ANS]cm. (Use cm for centimeters.)", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0001", "subject": "Geometry", "topic": "Shapes", "subtopic": "Perimeter", "level": "2", "keywords": [ "perimeter", "rectangle", "equation" ], "problem_v1": "A rectangle\u2019s perimeter is ${136\\ {\\rm ft}}$. Its length is ${4\\ {\\rm ft}}$ shorter than three times of its width. Use an equation to find the rectangle\u2019s length and width.\nIts width is [ANS]ft. Its length is [ANS].", "answer_v1": [ "18", "50" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rectangle\u2019s perimeter is ${98\\ {\\rm ft}}$. Its length is ${1\\ {\\rm ft}}$ shorter than four times of its width. Use an equation to find the rectangle\u2019s length and width.\nIts width is [ANS]ft. Its length is [ANS].", "answer_v2": [ "10", "39" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rectangle\u2019s perimeter is ${100\\ {\\rm ft}}$. Its length is ${2\\ {\\rm ft}}$ shorter than three times of its width. Use an equation to find the rectangle\u2019s length and width.\nIts width is [ANS]ft. Its length is [ANS].", "answer_v3": [ "13", "37" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0002", "subject": "Geometry", "topic": "Shapes", "subtopic": "Perimeter", "level": "1", "keywords": [], "problem_v1": "Max has a rectangular plot that is 80 square meters in area. He keeps his five goats, his twelve chickens and his brown and white horse in the plot which is enclosed on all four sides by a white picket fence. The length of the plot is 10 meters. So the total length of the white picket fence is [ANS] meters.", "answer_v1": [ "36" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Max has a rectangular plot that is 50 square meters in area. He keeps his five goats, his twelve chickens and his brown and white horse in the plot which is enclosed on all four sides by a white picket fence. The length of the plot is 10 meters. So the total length of the white picket fence is [ANS] meters.", "answer_v2": [ "30" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Max has a rectangular plot that is 60 square meters in area. He keeps his five goats, his twelve chickens and his brown and white horse in the plot which is enclosed on all four sides by a white picket fence. The length of the plot is 10 meters. So the total length of the white picket fence is [ANS] meters.", "answer_v3": [ "32" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0003", "subject": "Geometry", "topic": "Shapes", "subtopic": "Surface area", "level": "3", "keywords": [ "volume", "sphere" ], "problem_v1": "A sphere has a diameter of $6 cm$. What is the surface area? [ANS] $cm ^2$\nWhat is the volume? [ANS] $cm ^3$", "answer_v1": [ "113.097335529233", "113.097335529233" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A sphere has a diameter of $3.2 cm$. What is the surface area? [ANS] $cm ^2$\nWhat is the volume? [ANS] $cm ^3$", "answer_v2": [ "32.1699087727595", "17.1572846788051" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A sphere has a diameter of $4.2 cm$. What is the surface area? [ANS] $cm ^2$\nWhat is the volume? [ANS] $cm ^3$", "answer_v3": [ "55.4176944093239", "38.7923860865268" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0004", "subject": "Geometry", "topic": "Shapes", "subtopic": "Volume", "level": "2", "keywords": [ "volume", "rectangular", "prism" ], "problem_v1": "A rectangular prism\u2019s volume is ${9408\\ {\\rm ft^{3}}}$. The prism\u2019s base is a rectangle. The rectangle\u2019s base is ${28\\ {\\rm ft}}$ and the rectangle\u2019s height is ${16\\ {\\rm ft}}$.\nThis prism\u2019s height is [ANS]ft. (Use ft for feet.)", "answer_v1": [ "21" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rectangular prism\u2019s volume is ${3040\\ {\\rm ft^{3}}}$. The prism\u2019s base is a rectangle. The rectangle\u2019s base is ${20\\ {\\rm ft}}$ and the rectangle\u2019s height is ${19\\ {\\rm ft}}$.\nThis prism\u2019s height is [ANS]ft. (Use ft for feet.)", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rectangular prism\u2019s volume is ${4416\\ {\\rm ft^{3}}}$. The prism\u2019s base is a rectangle. The rectangle\u2019s base is ${23\\ {\\rm ft}}$ and the rectangle\u2019s height is ${16\\ {\\rm ft}}$.\nThis prism\u2019s height is [ANS]ft. (Use ft for feet.)", "answer_v3": [ "12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0005", "subject": "Geometry", "topic": "Shapes", "subtopic": "Volume", "level": "2", "keywords": [ "volume", "cylinder" ], "problem_v1": "A cylinder\u2019s height is ${7\\ {\\rm in}}$, and its volume is ${45\\ {\\rm in^{3}}}$.\nThis cylinder\u2019s radius is [ANS]in. Round your answer to the hundredths place. Use in for inches.", "answer_v1": [ "1.43" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A cylinder\u2019s height is ${10\\ {\\rm in}}$, and its volume is ${31\\ {\\rm in^{3}}}$.\nThis cylinder\u2019s radius is [ANS]in. Round your answer to the hundredths place. Use in for inches.", "answer_v2": [ "0.99" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A cylinder\u2019s height is ${7\\ {\\rm in}}$, and its volume is ${36\\ {\\rm in^{3}}}$.\nThis cylinder\u2019s radius is [ANS]in. Round your answer to the hundredths place. Use in for inches.", "answer_v3": [ "1.28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0006", "subject": "Geometry", "topic": "Shapes", "subtopic": "Volume", "level": "2", "keywords": [ "volume", "cube", "prism" ], "problem_v1": "A cube\u2019s side length is ${8\\ {\\rm cm}}$. Its volume is [ANS]cm^3. (Use cm^3 for cubic feet.)", "answer_v1": [ "512" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A cube\u2019s side length is ${2\\ {\\rm cm}}$. Its volume is [ANS]cm^3. (Use cm^3 for cubic feet.)", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A cube\u2019s side length is ${4\\ {\\rm cm}}$. Its volume is [ANS]cm^3. (Use cm^3 for cubic feet.)", "answer_v3": [ "64" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0007", "subject": "Geometry", "topic": "Shapes", "subtopic": "Volume", "level": "3", "keywords": [ "algebra", "linear equations", "volume", "pyramid" ], "problem_v1": "The volume of a pyramid is given by the equation V=\\frac{1}{3}Bh. Solve for $B.$ Answer: $B=$ [ANS]\nIf $V=180$ and $h=15,$ then what is the value of $B$? Answer: $B=$ [ANS]", "answer_v1": [ "\\frac{3V}{h}", "3*180/15" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The volume of a pyramid is given by the equation V=\\frac{1}{3}Bh. Solve for $B.$ Answer: $B=$ [ANS]\nIf $V=100$ and $h=25,$ then what is the value of $B$? Answer: $B=$ [ANS]", "answer_v2": [ "\\frac{3V}{h}", "3*100/25" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The volume of a pyramid is given by the equation V=\\frac{1}{3}Bh. Solve for $B.$ Answer: $B=$ [ANS]\nIf $V=120$ and $h=20,$ then what is the value of $B$? Answer: $B=$ [ANS]", "answer_v3": [ "\\frac{3V}{h}", "3*120/20" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0008", "subject": "Geometry", "topic": "Shapes", "subtopic": "Properties of shapes", "level": "3", "keywords": [], "problem_v1": "A square is divided into two triangles by one of its diagonals. Which of these words tells what kind of triangles are formed [ANS] A. Obtuse B. Acute C. Right\nAnother square is divided into two triangles by one of its diagonals. Which one of these words tells what kind of triangles are formed [ANS] A. Isoceles B. Equilateral C. Scalene", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "A square is divided into two triangles by one of its diagonals. Which of these words tells what kind of triangles are formed [ANS] A. Right B. Acute C. Obtuse\nAnother square is divided into two triangles by one of its diagonals. Which one of these words tells what kind of triangles are formed [ANS] A. Isoceles B. Scalene C. Equilateral", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "A square is divided into two triangles by one of its diagonals. Which of these words tells what kind of triangles are formed [ANS] A. Obtuse B. Right C. Acute\nAnother square is divided into two triangles by one of its diagonals. Which one of these words tells what kind of triangles are formed [ANS] A. Equilateral B. Isoceles C. Scalene", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Geometry_0009", "subject": "Geometry", "topic": "Shapes", "subtopic": "Properties of shapes", "level": "3", "keywords": [], "problem_v1": "Which shape is possible? [ANS] A. a rhombus with sides that measure 4 cm, 4 cm, 8cm, 8cm. B. a rhombus with 4 acute angles. C. a parallelogram with 4 angles of equal measure. D. a parallelogram with sides that measure 2 cm, 4cm, 6 cm, 8 cm.", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Which shape is possible? [ANS] A. a parallelogram with 4 angles of equal measure. B. a rhombus with 4 acute angles. C. a rhombus with sides that measure 4 cm, 4 cm, 8cm, 8cm. D. a parallelogram with sides that measure 2 cm, 4cm, 6 cm, 8 cm.", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Which shape is possible? [ANS] A. a rhombus with sides that measure 4 cm, 4 cm, 8cm, 8cm. B. a parallelogram with 4 angles of equal measure. C. a rhombus with 4 acute angles. D. a parallelogram with sides that measure 2 cm, 4cm, 6 cm, 8 cm.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Geometry_0010", "subject": "Geometry", "topic": "Shapes", "subtopic": "Properties of shapes", "level": "4", "keywords": [ "quadrilateral" ], "problem_v1": "QUADRILATERAL Which statements are true? [ANS] A. Every rhombus is a rectangle B. Every parallelogram is a rhombus C. Every rhombus is a square D. Every deltoid is a parallelogram E. Every rectangle is a parallelogram F. Every parallelogram is a trapezoid", "answer_v1": [ "EF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "QUADRILATERAL Which statements are true? [ANS] A. Every rhombus is a trapezoid B. Every rectangle is a parallelogram C. Every rhombus is a rectangle D. Every rectangle is a quadrilateral E. Every parallelogram is a square F. Every square is a trapezoid", "answer_v2": [ "ABDF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "QUADRILATERAL Which statements are true? [ANS] A. Every deltoid is a rhombus B. Every parallelogram is a quadrilateral C. Every square is a rectangle D. Every trapezoid is a quadrilateral E. Every trapezoid is a parallelogram F. Every trapezoid is a quadrilateral", "answer_v3": [ "BCDF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Geometry_0011", "subject": "Geometry", "topic": "Circle geometry", "subtopic": "Circumference and area", "level": "2", "keywords": [ "area", "circumference", "circle" ], "problem_v1": "A circle\u2019s radius is ${8\\ {\\rm m}}$.\nThis circle\u2019s circumference, in terms of $\\pi$, is [ANS]m. (Type pi for $\\pi$.)\nThis circle\u2019s circumference, rounded to the hundredth place, is [ANS].\nThis circle\u2019s area, in terms of $\\pi$, is [ANS]. (Type pi for $\\pi$.)\nThis circle\u2019s area, rounded to the hundredth place, is [ANS]. (Use m for meters and m^2 for square meters.)", "answer_v1": [ "16*pi", "50.27", "64*pi", "201.06" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A circle\u2019s radius is ${2\\ {\\rm m}}$.\nThis circle\u2019s circumference, in terms of $\\pi$, is [ANS]m. (Type pi for $\\pi$.)\nThis circle\u2019s circumference, rounded to the hundredth place, is [ANS].\nThis circle\u2019s area, in terms of $\\pi$, is [ANS]. (Type pi for $\\pi$.)\nThis circle\u2019s area, rounded to the hundredth place, is [ANS]. (Use m for meters and m^2 for square meters.)", "answer_v2": [ "4*pi", "12.57", "4*pi", "12.57" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A circle\u2019s radius is ${4\\ {\\rm m}}$.\nThis circle\u2019s circumference, in terms of $\\pi$, is [ANS]m. (Type pi for $\\pi$.)\nThis circle\u2019s circumference, rounded to the hundredth place, is [ANS].\nThis circle\u2019s area, in terms of $\\pi$, is [ANS]. (Type pi for $\\pi$.)\nThis circle\u2019s area, rounded to the hundredth place, is [ANS]. (Use m for meters and m^2 for square meters.)", "answer_v3": [ "8*pi", "25.13", "16*pi", "50.27" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0012", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "vector" ], "problem_v1": "Calculate: $\\left<4,3\\right>+\\left<3,3\\right>=$ [ANS]", "answer_v1": [ "(7, 6)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Calculate: $\\left<1,5\\right>+\\left<1,2\\right>=$ [ANS]", "answer_v2": [ "(2, 7)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Calculate: $\\left<2,4\\right>+\\left<2,3\\right>=$ [ANS]", "answer_v3": [ "(4, 7)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0013", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "vector" ], "problem_v1": "If $P=(2,1)$ and $Q=(3,3)$, find the components of $\\vec{PQ}$ $\\vec{PQ}=$ [ANS]", "answer_v1": [ "(1, 2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "If $P=(-4,4)$ and $Q=(-8,2)$, find the components of $\\vec{PQ}$ $\\vec{PQ}=$ [ANS]", "answer_v2": [ "(-4, -2)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "If $P=(-2,1)$ and $Q=(-4,2)$, find the components of $\\vec{PQ}$ $\\vec{PQ}=$ [ANS]", "answer_v3": [ "(-2, 1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0014", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vector" ], "problem_v1": "Suppose that $ABCD$ is a parallelogram, and $A=(-2,3),\\qquad B=(1,b),\\qquad C=(3,3),\\qquad D=(a, 0)$\nWhat are the values of $a$ and $b$? $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "0", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $ABCD$ is a parallelogram, and $A=(-5,5),\\qquad B=(-4,b),\\qquad C=(0,4),\\qquad D=(a, 2)$\nWhat are the values of $a$ and $b$? $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-1", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $ABCD$ is a parallelogram, and $A=(-4,4),\\qquad B=(-2,b),\\qquad C=(-1,4),\\qquad D=(a, 1)$\nWhat are the values of $a$ and $b$? $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-3", "7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0015", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vector" ], "problem_v1": "What is the terminal point of the vector $\\mathbf{a}=\\left<4,3\\right>$ based at $P=\\left(4,4\\right)$?\nAnswer: [ANS]", "answer_v1": [ "(8,7)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "What is the terminal point of the vector $\\mathbf{a}=\\left<1,5\\right>$ based at $P=\\left(1,2\\right)$?\nAnswer: [ANS]", "answer_v2": [ "(2,7)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "What is the terminal point of the vector $\\mathbf{a}=\\left<2,4\\right>$ based at $P=\\left(2,3\\right)$?\nAnswer: [ANS]", "answer_v3": [ "(4,7)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0016", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vector" ], "problem_v1": "Let $R=\\left(3,1\\right)$. Find the point $P$ such that $\\vec{PR}$ has components $\\left<1,2\\right>$.\n$P=$ [ANS]", "answer_v1": [ "(2,-1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Let $R=\\left(-5,5\\right)$. Find the point $P$ such that $\\vec{PR}$ has components $\\left<-2,-1\\right>$.\n$P=$ [ANS]", "answer_v2": [ "(-3,6)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Let $R=\\left(-2,1\\right)$. Find the point $P$ such that $\\vec{PR}$ has components $\\left<-2,0\\right>$.\n$P=$ [ANS]", "answer_v3": [ "(0,1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0017", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "$\\mathbf{v}=\\left<8,14,21\\right>$ Which of the following vectors is parallel to $\\mathbf{v}$? [ANS] A. $<8,-14,-21>$ B. $<-8,-14,-21>$ C. $<8,7,5.25>$ D. $<-14,21,-8>$ E. $<21,14,8>$\nWhich of the following points in the same direction as $\\mathbf{v}$? [ANS] A. $<16,28,42>$ B. $<8,-14,-21>$ C. $<-14,21,-8>$ D. $<-8,-14,-21>$ E. $<14,21,8>$", "answer_v1": [ "B", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "$\\mathbf{v}=\\left<1,11,13\\right>$ Which of the following vectors is parallel to $\\mathbf{v}$? [ANS] A. $<1,5.5,3.25>$ B. $<1,-11,-13>$ C. $<-1,-11,-13>$ D. $<13,11,1>$ E. $<-11,13,-1>$\nWhich of the following points in the same direction as $\\mathbf{v}$? [ANS] A. $<1,-11,-13>$ B. $<-11,13,-1>$ C. $<2,22,26>$ D. $<11,13,1>$ E. $<-1,-11,-13>$", "answer_v2": [ "C", "C" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "$\\mathbf{v}=\\left<4,11,14\\right>$ Which of the following vectors is parallel to $\\mathbf{v}$? [ANS] A. $<4,-11,-14>$ B. $<4,5.5,3.5>$ C. $<14,11,4>$ D. $<-4,-11,-14>$ E. $<-11,14,-4>$\nWhich of the following points in the same direction as $\\mathbf{v}$? [ANS] A. $<-4,-11,-14>$ B. $<4,-11,-14>$ C. $<-11,14,-4>$ D. $<8,22,28>$ E. $<11,14,4>$", "answer_v3": [ "D", "D" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Geometry_0018", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Find the point P such that $\\mathbf{v}=\\overrightarrow{PQ}$ has the components $\\left<-3.5,-10,-10.5\\right>$. $Q=\\left(4.5,-4,-3.5\\right)$ $P$=[ANS]", "answer_v1": [ "(8,6,7)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the point P such that $\\mathbf{v}=\\overrightarrow{PQ}$ has the components $\\left<-4.25,-1,-6\\right>$. $Q=\\left(-3.25,9,-4\\right)$ $P$=[ANS]", "answer_v2": [ "(1,10,2)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the point P such that $\\mathbf{v}=\\overrightarrow{PQ}$ has the components $\\left<-3,-13,-6\\right>$. $Q=\\left(1,-6,-3\\right)$ $P$=[ANS]", "answer_v3": [ "(4,7,3)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0019", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Calculate the linear combination: $4 \\left<1,2,-2\\right>+3 \\left<-2,1,1\\right>$=[ANS]", "answer_v1": [ "(-2, 11, -5;)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Calculate the linear combination: $1 \\left<-4,-2,5\\right>+4 \\left<-2,-3,-2\\right>$=[ANS]", "answer_v2": [ "(-12, -14, -3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Calculate the linear combination: $2 \\left<-2,1,-3\\right>+3 \\left<-2,3,5\\right>$=[ANS]", "answer_v3": [ "(-10, 11, 9)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0020", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Find the vector $\\mathbf{v}=\\overrightarrow{PQ}$. $P=\\left(8,6,7\\right)$ $Q=\\left(8,4,4\\right)$ $\\mathbf{v}$=[ANS]", "answer_v1": [ "(0, -2, -3)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the vector $\\mathbf{v}=\\overrightarrow{PQ}$. $P=\\left(1,10,2\\right)$ $Q=\\left(4,10,4\\right)$ $\\mathbf{v}$=[ANS]", "answer_v2": [ "(3, 0, 2;)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the vector $\\mathbf{v}=\\overrightarrow{PQ}$. $P=\\left(4,7,3\\right)$ $Q=\\left(6,3,4\\right)$ $\\mathbf{v}$=[ANS]", "answer_v3": [ "(2, -4, 1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0021", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "Vector" ], "problem_v1": "Suppose $\\overline{u}=\\left<3,1\\right>$ and $\\overline{v}=\\left<1,2\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\overline{u}+\\overline{v} &=& [ANS] \\\\\\hline \\overline{u}-\\overline{v} &=& [ANS] \\\\\\hline \\overline{v}-\\overline{u} &=& [ANS] \\\\\\hline 4\\overline{u} &=& [ANS] \\\\\\hline-\\frac{1}{5}\\overline{v} &=& [ANS] \\\\\\hline 5\\overline{u}-6\\overline{v} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v1": [ "(4,3)", "(2,-1)", "(-2,1)", "(12,4)", "(-0.2,-0.4)", "(9,-7)" ], "answer_type_v1": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose $\\overline{u}=\\left<-5,5\\right>$ and $\\overline{v}=\\left<-4,-2\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\overline{u}+\\overline{v} &=& [ANS] \\\\\\hline \\overline{u}-\\overline{v} &=& [ANS] \\\\\\hline \\overline{v}-\\overline{u} &=& [ANS] \\\\\\hline 8\\overline{u} &=& [ANS] \\\\\\hline-\\frac{1}{4}\\overline{v} &=& [ANS] \\\\\\hline 3\\overline{u}-4\\overline{v} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v2": [ "(-9,3)", "(-1,7)", "(1,-7)", "(-40,40)", "(1,0.5)", "(1,23)" ], "answer_type_v2": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose $\\overline{u}=\\left<-2,1\\right>$ and $\\overline{v}=\\left<-2,1\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\overline{u}+\\overline{v} &=& [ANS] \\\\\\hline \\overline{u}-\\overline{v} &=& [ANS] \\\\\\hline \\overline{v}-\\overline{u} &=& [ANS] \\\\\\hline 3\\overline{u} &=& [ANS] \\\\\\hline-\\frac{1}{4}\\overline{v} &=& [ANS] \\\\\\hline 7\\overline{u}-8\\overline{v} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v3": [ "(-4,2)", "(0,0)", "(0,0)", "(-6,3)", "(0.5,-0.25)", "(2,-1)" ], "answer_type_v3": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Geometry_0022", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "Vector" ], "problem_v1": "Let $\\overline{u}=\\left<3,1\\right>$, $\\overline{v}=\\left<1,2\\right>$, and $\\overline{w}=\\left<-2,-2\\right>$. Find the vector $\\overline{x}$ that satisfies 7 \\overline{u}-\\overline{v}+\\overline{x}=7 \\overline{x}+\\overline{w}. In this case, $\\overline{x}=$ [ANS].", "answer_v1": [ "(3.66667,1.16667)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Let $\\overline{u}=\\left<-5,5\\right>$, $\\overline{v}=\\left<-4,-2\\right>$, and $\\overline{w}=\\left<5,-2\\right>$. Find the vector $\\overline{x}$ that satisfies 3 \\overline{u}-\\overline{v}+\\overline{x}=4 \\overline{x}+\\overline{w}. In this case, $\\overline{x}=$ [ANS].", "answer_v2": [ "(-5.33333,6.33333)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Let $\\overline{u}=\\left<-2,1\\right>$, $\\overline{v}=\\left<-2,1\\right>$, and $\\overline{w}=\\left<-3,-2\\right>$. Find the vector $\\overline{x}$ that satisfies 9 \\overline{u}-\\overline{v}+\\overline{x}=10 \\overline{x}+\\overline{w}. In this case, $\\overline{x}=$ [ANS].", "answer_v3": [ "(-1.44444,1.11111)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0023", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "Vector", "Equal" ], "problem_v1": "For what values of $t$ and $s$ does the equality \\left<5t+2,2s+4t\\right>=\\left<3s-t,-\\left(2+2t\\right)\\right> hold true?\n$t$=[ANS] and $s$=[ANS]\nAt these values, the resulting vector is [ANS].", "answer_v1": [ "-0.333333", "0", "(0.333333,-1.33333)" ], "answer_type_v1": [ "NV", "NV", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "For what values of $t$ and $s$ does the equality \\left<-\\left(7t+1\\right),2s+t\\right>=\\left hold true?\n$t$=[ANS] and $s$=[ANS]\nAt these values, the resulting vector is [ANS].", "answer_v2": [ "-0.333333", "1", "(1.33333,1.66667)" ], "answer_type_v2": [ "NV", "NV", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "For what values of $t$ and $s$ does the equality \\left=\\left<4s-5t,-\\left(1+2t\\right)\\right> hold true?\n$t$=[ANS] and $s$=[ANS]\nAt these values, the resulting vector is [ANS].", "answer_v3": [ "0.166667", "-0.5", "(-2.83333,-1.33333)" ], "answer_type_v3": [ "NV", "NV", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0024", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "Vector", "Equal" ], "problem_v1": "For what value(s) of $t$ does the equality \\textstyle \\left=\\left<0,7\\right> hold true?\n$t$=[ANS]. (If there are more than one value, separate them by commas.) (If there are more than one value, separate them by commas.)", "answer_v1": [ "(4, -4)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "For what value(s) of $t$ does the equality \\textstyle \\left=\\left<0,6\\right> hold true?\n$t$=[ANS]. (If there are more than one value, separate them by commas.) (If there are more than one value, separate them by commas.)", "answer_v2": [ "(1, -1)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "For what value(s) of $t$ does the equality \\textstyle \\left=\\left<0,6\\right> hold true?\n$t$=[ANS]. (If there are more than one value, separate them by commas.) (If there are more than one value, separate them by commas.)", "answer_v3": [ "(2, -2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Geometry_0026", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "multivariable", "functions" ], "problem_v1": "A cube is located such that its bottom four corners have the coordinates $\\left(-2,-2,-1\\right)$, $\\left(-2,4,-1\\right)$, $\\left(4,-2,-1\\right)$ and $\\left(4,4,-1\\right)$. Give the coordinates of the center of the cube. center=[ANS]", "answer_v1": [ "(1,1,2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "A cube is located such that its top four corners have the coordinates $\\left(-5,-4,3\\right)$, $\\left(-5,4,3\\right)$, $\\left(3,-4,3\\right)$ and $\\left(3,4,3\\right)$. Give the coordinates of the center of the cube. center=[ANS]", "answer_v2": [ "(-1,0,-1)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "A cube is located such that its top four corners have the coordinates $\\left(-4,-3,1\\right)$, $\\left(-4,4,1\\right)$, $\\left(3,-3,1\\right)$ and $\\left(3,4,1\\right)$. Give the coordinates of the center of the cube. center=[ANS]", "answer_v3": [ "(-0.5,0.5,-2.5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0027", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "For each of the following, perform the indicated computation.\n(a) $(5\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}+3\\,\\mathit{\\vec k})-(5\\,\\mathit{\\vec i}-4\\,\\mathit{\\vec j}-4\\,\\mathit{\\vec k})=$ [ANS]\n(b) $(\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}-3\\,\\mathit{\\vec k})-3(3\\,\\mathit{\\vec i}-5\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k})=$ [ANS]", "answer_v1": [ "6j+7k", "-8i+17j+3k" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "For each of the following, perform the indicated computation.\n(a) $(-9\\,\\mathit{\\vec i}+9\\,\\mathit{\\vec j}-7\\,\\mathit{\\vec k})-(-3\\,\\mathit{\\vec i}+9\\,\\mathit{\\vec j}-4\\,\\mathit{\\vec k})=$ [ANS]\n(b) $(-7\\,\\mathit{\\vec i}-4\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k})-4(-9\\,\\mathit{\\vec i}+3\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k})=$ [ANS]", "answer_v2": [ "-6i-3k", "29i-16j+5k" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "For each of the following, perform the indicated computation.\n(a) $(-4\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}-5\\,\\mathit{\\vec k})-(\\,\\mathit{\\vec i}-6\\,\\mathit{\\vec j}-3\\,\\mathit{\\vec k})=$ [ANS]\n(b) $(6\\,\\mathit{\\vec i}+9\\,\\mathit{\\vec j}+8\\,\\mathit{\\vec k})-2(-6\\,\\mathit{\\vec i}-4\\,\\mathit{\\vec j}-5\\,\\mathit{\\vec k})=$ [ANS]", "answer_v3": [ "-5i+8j-2k", "18i+17j+18k" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0028", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "A truck is traveling due north at $65$ km/hr approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at $60$ km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car. displacement $\\vec d=$ [ANS]", "answer_v1": [ "-60i-65j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A truck is traveling due north at $35$ km/hr approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at $75$ km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car. displacement $\\vec d=$ [ANS]", "answer_v2": [ "-75i-35j" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A truck is traveling due north at $45$ km/hr approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at $60$ km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car. displacement $\\vec d=$ [ANS]", "answer_v3": [ "-60i-45j" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0029", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "Resolve the following vectors into components:\n(a) The vector $\\vec v$ in 2-space of length 5 pointing up at an angle of $2\\pi/3$ measured from the positive $x$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$ (b) The vector $\\vec w$ in 3-space of length 3 lying in the $yz$-plane pointing upward at an angle of $3\\pi/4$ measured from the positive $y$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$+[ANS] $\\vec k$", "answer_v1": [ "5*-0.5", "5*0.866025", "0", "3*-0.707107", "3*0.707107" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Resolve the following vectors into components:\n(a) The vector $\\vec v$ in 2-space of length 3 pointing up at an angle of $\\pi/6$ measured from the positive $x$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$ (b) The vector $\\vec w$ in 3-space of length 1 lying in the $xz$-plane pointing upward at an angle of $5\\pi/6$ measured from the positive $x$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$+[ANS] $\\vec k$", "answer_v2": [ "3*0.866025", "3*0.5", "1*-0.866025", "0", "1*0.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Resolve the following vectors into components:\n(a) The vector $\\vec v$ in 2-space of length 3 pointing up at an angle of $\\pi/4$ measured from the positive $x$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$ (b) The vector $\\vec w$ in 3-space of length 1 lying in the $yz$-plane pointing upward at an angle of $3\\pi/4$ measured from the positive $y$-axis. $\\vec v=$ [ANS] $\\vec i$+[ANS] $\\vec j$+[ANS] $\\vec k$", "answer_v3": [ "3*0.707107", "3*0.707107", "0", "1*-0.707107", "1*0.707107" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Geometry_0030", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors", "force" ], "problem_v1": "There are five students in a class. Their scores on the midterm (out of 100) are given by the vector $\\vec v=<88,84,74,82,76>$. Their scores on the final (out of 100) are given by $\\vec w=<89,86,72,83,78>$. If the final counts twice as much as the midterm, find a vector $\\vec z$ giving the total scores (as a percentage) of the students. $\\vec z=$ [ANS]", "answer_v1": [ "(88.6667,85.3333,72.6667,82.6667,77.3333)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "There are five students in a class. Their scores on the midterm (out of 100) are given by the vector $\\vec v=<67,69,94,70,82>$. Their scores on the final (out of 100) are given by $\\vec w=<72,67,92,68,77>$. If the final counts twice as much as the midterm, find a vector $\\vec z$ giving the total scores (as a percentage) of the students. $\\vec z=$ [ANS]", "answer_v2": [ "(70.3333,67.6667,92.6667,68.6667,78.6667)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "There are five students in a class. Their scores on the midterm (out of 100) are given by the vector $\\vec v=<74,73,71,90,92>$. Their scores on the final (out of 100) are given by $\\vec w=<75,74,69,95,89>$. If the final counts twice as much as the midterm, find a vector $\\vec z$ giving the total scores (as a percentage) of the students. $\\vec z=$ [ANS]", "answer_v3": [ "(74.6667,73.6667,69.6667,93.3333,90)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0031", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors", "force" ], "problem_v1": "Give the velocity vector for wind blowing at 15 km/hr toward the southwest. (Assume north is the positive $y$-direction.) $\\vec v=$ [ANS]", "answer_v1": [ "-10.6066i-10.6066j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Give the velocity vector for wind blowing at 25 km/hr toward the northwest. (Assume north is the positive $y$-direction.) $\\vec v=$ [ANS]", "answer_v2": [ "-17.6777i+17.6777j" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Give the velocity vector for wind blowing at 20 km/hr toward the northeast. (Assume north is the positive $y$-direction.) $\\vec v=$ [ANS]", "answer_v3": [ "14.1421i+14.1421j" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0032", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "4", "keywords": [ "vectors", "force" ], "problem_v1": "Two forces, represented by the vectors $\\vec F_1=6\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}$ and $\\vec F_2=4\\,\\mathit{\\vec i}+6\\,\\mathit{\\vec j}$, are acting on an object. Give a vector $\\vec F_3$ representing the force that must be applied to the object if it is to remain stationary. $\\vec F_3=$ [ANS]", "answer_v1": [ "-10i-8j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Two forces, represented by the vectors $\\vec F_1=-10\\,\\mathit{\\vec i}+10\\,\\mathit{\\vec j}$ and $\\vec F_2=-3\\,\\mathit{\\vec i}+10\\,\\mathit{\\vec j}$, are acting on an object. Give a vector $\\vec F_3$ representing the force that must be applied to the object if it is to remain stationary. $\\vec F_3=$ [ANS]", "answer_v2": [ "13i-20j" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Two forces, represented by the vectors $\\vec F_1=-4\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}$ and $\\vec F_2=-\\,\\mathit{\\vec i}+3\\,\\mathit{\\vec j}$, are acting on an object. Give a vector $\\vec F_3$ representing the force that must be applied to the object if it is to remain stationary. $\\vec F_3=$ [ANS]", "answer_v3": [ "5i-5j" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0033", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "4", "keywords": [ "vectors", "force" ], "problem_v1": "A large ship is being towed by two tugs. The larger tug exerts a force which is 30 percent greater than the smaller tug, and at an angle of 30 degrees south of west. Which direction must the smaller tug pull to ensure that the ship travels due west? It should pull at an angle of [ANS] degrees [ANS] of [ANS].", "answer_v1": [ "180/pi*asin(1.3*sin(30*pi/180))", "north", "west" ], "answer_type_v1": [ "NV", "MCS", "MCS" ], "options_v1": [ [], [ "north", "south" ], [ "east", "west" ] ], "problem_v2": "A large ship is being towed by two tugs. The larger tug exerts a force which is 15 percent greater than the smaller tug, and at an angle of 35 degrees north of east. Which direction must the smaller tug pull to ensure that the ship travels due east? It should pull at an angle of [ANS] degrees [ANS] of [ANS].", "answer_v2": [ "180/pi*asin(1.15*sin(35*pi/180))", "south", "east" ], "answer_type_v2": [ "NV", "MCS", "MCS" ], "options_v2": [ [], [ "north", "south" ], [ "east", "west" ] ], "problem_v3": "A large ship is being towed by two tugs. The larger tug exerts a force which is 20 percent greater than the smaller tug, and at an angle of 30 degrees north of west. Which direction must the smaller tug pull to ensure that the ship travels due west? It should pull at an angle of [ANS] degrees [ANS] of [ANS].", "answer_v3": [ "180/pi*asin(1.2*sin(30*pi/180))", "south", "west" ], "answer_type_v3": [ "NV", "MCS", "MCS" ], "options_v3": [ [], [ "north", "south" ], [ "east", "west" ] ] }, { "id": "Geometry_0034", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "1", "keywords": [ "vectors" ], "problem_v1": "Find the vector in $\\mathbb{R}^3$ from point $A=\\left(x,y,z\\right)$ to $B=\\left(5,2,2\\right)$.\n$\\overrightarrow{AB}=$ [ANS]", "answer_v1": [ "(5-x,2-y,2-z)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the vector in $\\mathbb{R}^3$ from point $A=\\left(x,y,z\\right)$ to $B=\\left(-8,8,-7\\right)$.\n$\\overrightarrow{AB}=$ [ANS]", "answer_v2": [ "(-(8+x),8-y,-(7+z))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the vector in $\\mathbb{R}^3$ from point $A=\\left(x,y,z\\right)$ to $B=\\left(-4,2,-4\\right)$.\n$\\overrightarrow{AB}=$ [ANS]", "answer_v3": [ "(-(4+x),2-y,-(4+z))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0035", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "1", "keywords": [ "vectors" ], "problem_v1": "Suppose $B=\\left(4,-4,-3\\right)$ and $\\overrightarrow{AB}=\\left<-1,-6,-5\\right>$. Then\n$A=$ [ANS]", "answer_v1": [ "(5,2,2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Suppose $B=\\left(-6,-3,1\\right)$ and $\\overrightarrow{AB}=\\left<2,-11,8\\right>$. Then\n$A=$ [ANS]", "answer_v2": [ "(-8,8,-7)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Suppose $B=\\left(1,-6,-3\\right)$ and $\\overrightarrow{AB}=\\left<5,-8,1\\right>$. Then\n$A=$ [ANS]", "answer_v3": [ "(-4,2,-4)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0036", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "1", "keywords": [ "vectors" ], "problem_v1": "Find a representation of the vector $\\overrightarrow{AB}=\\left<-3,2\\right>$ in $\\mathbb{R}^2$ by giving appropriate values for the points $A$ and $B$ such that neither $A$ nor $B$ is the origin.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v1": [ "(5,2)", "(2,4)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Find a representation of the vector $\\overrightarrow{AB}=\\left<1,-11\\right>$ in $\\mathbb{R}^2$ by giving appropriate values for the points $A$ and $B$ such that neither $A$ nor $B$ is the origin.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v2": [ "(-8,8)", "(-7,-3)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find a representation of the vector $\\overrightarrow{AB}=\\left<0,-1\\right>$ in $\\mathbb{R}^2$ by giving appropriate values for the points $A$ and $B$ such that neither $A$ nor $B$ is the origin.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v3": [ "(-4,2)", "(-4,1)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0037", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "Vector", "Length" ], "problem_v1": "Suppose a car is traveling at 65 km/hr, and that the positive y-axis is north and the positive x-axis is east. Resolve the car's velocity vector (in 2-space) into components if the car is traveling in each of the following directions. The units are km/hr, but you do not need to enter units. you do not need to enter units.\nEast: [ANS]. South: [ANS]. Southeast: [ANS]. Northwest: [ANS].", "answer_v1": [ "(65,0)", "(0,-65)", "(45.9619,-45.9619)", "(-45.9619,45.9619)" ], "answer_type_v1": [ "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose a car is traveling at 50 km/hr, and that the positive y-axis is north and the positive x-axis is east. Resolve the car's velocity vector (in 2-space) into components if the car is traveling in each of the following directions. The units are km/hr, but you do not need to enter units. you do not need to enter units.\nEast: [ANS]. South: [ANS]. Southeast: [ANS]. Northwest: [ANS].", "answer_v2": [ "(50,0)", "(0,-50)", "(35.3553,-35.3553)", "(-35.3553,35.3553)" ], "answer_type_v2": [ "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose a car is traveling at 55 km/hr, and that the positive y-axis is north and the positive x-axis is east. Resolve the car's velocity vector (in 2-space) into components if the car is traveling in each of the following directions. The units are km/hr, but you do not need to enter units. you do not need to enter units.\nEast: [ANS]. South: [ANS]. Southeast: [ANS]. Northwest: [ANS].", "answer_v3": [ "(55,0)", "(0,-55)", "(38.8909,-38.8909)", "(-38.8909,38.8909)" ], "answer_type_v3": [ "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0038", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "2", "keywords": [ "Vector", "Length" ], "problem_v1": "Which of the following vectors are parallel?\nPlease select the two vectors that are parallel. [ANS] A. $<4,4,0>$ B. $<5,5,10>$ C. $<6,-9,0>$ D. $<-20,-20,-40>$ E. $<20,5,0>$ F. $<3,3,-3>$\nPlease select the two vectors that are parallel. [ANS] A. $<-30,0,60>$ B. $<16,4,-12>$ C. $<-4,6,6>$ D. $<-15,-9,0>$ E. $<15,15,5>$ F. $<-6,0,12>$", "answer_v1": [ "BD", "AF" ], "answer_type_v1": [ "MCM", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following vectors are parallel?\nPlease select the two vectors that are parallel. [ANS] A. $<-20,16,8>$ B. $<-9,-6,3>$ C. $<15,15,-5>$ D. $<10,-8,-4>$ E. $<4,-2,6>$ F. $<10,-5,20>$\nPlease select the two vectors that are parallel. [ANS] A. $<-20,0,-40>$ B. $<-4,0,-8>$ C. $<-3,6,15>$ D. $<0,12,-15>$ E. $<4,-12,-20>$ F. $<20,8,-12>$", "answer_v2": [ "AD", "AB" ], "answer_type_v2": [ "MCM", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following vectors are parallel?\nPlease select the two vectors that are parallel. [ANS] A. $<10,-10,-10>$ B. $<-4,-6,-10>$ C. $<3,-6,3>$ D. $<-15,-9,-3>$ E. $<9,15,12>$ F. $<-15,30,-15>$\nPlease select the two vectors that are parallel. [ANS] A. $<20,12,-16>$ B. $<0,12,0>$ C. $<16,8,0>$ D. $<20,12,-12>$ E. $<60,36,48>$ F. $<15,9,12>$", "answer_v3": [ "CF", "EF" ], "answer_type_v3": [ "MCM", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Geometry_0039", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors" ], "problem_v1": "If $A=(5,2)$ then what is the vector from the origin to $A$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$ And what is the vector from $A$ to the origin? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v1": [ "5", "2", "-5", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $A=(-9,9)$ then what is the vector from the origin to $A$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$ And what is the vector from $A$ to the origin? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v2": [ "-9", "9", "9", "-9" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $A=(-4,2)$ then what is the vector from the origin to $A$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$ And what is the vector from $A$ to the origin? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v3": [ "-4", "2", "4", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0040", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Consider the (approximate) unit vector $\\langle 0.04,-1 \\rangle$. Give another unit vector different from it, but still parallel to it. Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v1": [ "-0.04", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the (approximate) unit vector $\\langle 0.82,0.57 \\rangle$. Give another unit vector different from it, but still parallel to it. Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v2": [ "-0.82", "-0.57" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the (approximate) unit vector $\\langle-0.45,0.89 \\rangle$. Give another unit vector different from it, but still parallel to it. Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v3": [ "0.45", "-0.89" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0041", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Assume we wish to make an animation that moves an object from the point $A=(10,3)$ to the point $B=(5,9)$. What sum $P+\\vec v$ is involved in this animation? Answer: $($ [ANS], [ANS] $)~+~\\langle$ [ANS], [ANS] $\\rangle$ Compute $P+\\frac12\\vec v$ to find where the object would be half way along its path. Answer: $($ [ANS], [ANS] $)$ Where is it when it is $90\\%$ of the way along? Answer: $($ [ANS], [ANS] $)$", "answer_v1": [ "10", "3", "-5", "6", "7.5", "6", "5.5", "8.4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Assume we wish to make an animation that moves an object from the point $A=(-17,18)$ to the point $B=(-14,-7)$. What sum $P+\\vec v$ is involved in this animation? Answer: $($ [ANS], [ANS] $)~+~\\langle$ [ANS], [ANS] $\\rangle$ Compute $P+\\frac12\\vec v$ to find where the object would be half way along its path. Answer: $($ [ANS], [ANS] $)$ Where is it when it is $90\\%$ of the way along? Answer: $($ [ANS], [ANS] $)$", "answer_v2": [ "-17", "18", "3", "-25", "-15.5", "5.5", "-14.3", "-4.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Assume we wish to make an animation that moves an object from the point $A=(-8,4)$ to the point $B=(-9,2)$. What sum $P+\\vec v$ is involved in this animation? Answer: $($ [ANS], [ANS] $)~+~\\langle$ [ANS], [ANS] $\\rangle$ Compute $P+\\frac12\\vec v$ to find where the object would be half way along its path. Answer: $($ [ANS], [ANS] $)$ Where is it when it is $90\\%$ of the way along? Answer: $($ [ANS], [ANS] $)$", "answer_v3": [ "-8", "4", "-1", "-2", "-8.5", "3", "-8.9", "2.2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0042", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors" ], "problem_v1": "What is the vector between the point $({5a},{2b})$ and the point $({3a},{5b})$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v1": [ "-2*a", "3*b" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "What is the vector between the point $({-9a},{9b})$ and the point $({-7a},{-3b})$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v2": [ "2*a", "-12*b" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "What is the vector between the point $({-4a},{2b})$ and the point $({-5a},{b})$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v3": [ "-a", "-b" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0043", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Vectors and vector arithmetic", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Let $\\vec v$ be the vector $\\langle {5a},{2b},{3c} \\rangle$ and $A$ be the point $({5a},{-4b},{-4c})$. What is the endoint of $\\vec v$ if it starts at $A$? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "10*a", "-2*b", "-c" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\vec v$ be the vector $\\langle {-9a},{9b},{-7c} \\rangle$ and $A$ be the point $({-3a},{9b},{-4c})$. What is the endoint of $\\vec v$ if it starts at $A$? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "-12*a", "18*b", "-11*c" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\vec v$ be the vector $\\langle {-4a},{2b},{-5c} \\rangle$ and $A$ be the point $({a},{-6b},{-3c})$. What is the endoint of $\\vec v$ if it starts at $A$? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "-3*a", "-4*b", "-8*c" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0044", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vector" ], "problem_v1": "Let $R=(3,1)$. Calculate the length of $\\vec{OR}$. [ANS]", "answer_v1": [ "3.16228" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $R=(-5,5)$. Calculate the length of $\\vec{OR}$. [ANS]", "answer_v2": [ "7.07107" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $R=(-2,1)$. Calculate the length of $\\vec{OR}$. [ANS]", "answer_v3": [ "2.23607" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0046", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vector" ], "problem_v1": "Find the unit vector in the direction opposite to ${\\bf v}=\\left<3,1\\right>$. [ANS]", "answer_v1": [ "(-0.948683,-0.316228)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the unit vector in the direction opposite to ${\\bf v}=\\left<-5,5\\right>$. [ANS]", "answer_v2": [ "(0.707107,-0.707107)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the unit vector in the direction opposite to ${\\bf v}=\\left<-2,1\\right>$. [ANS]", "answer_v3": [ "(0.894427,-0.447214)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0047", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "$\\| v \\|=3$ $\\| w \\|=4$ The angle between $v$ and $w$ is 2.4 radians. Given this information, calculate the following:\n(a) $v \\cdotp w$=[ANS]\n(b) $\\| 3 v+2 w \\|=$ [ANS]\n(c) $\\| 2 v-2 w \\|=$ [ANS]", "answer_v1": [ "-8.84872", "6.23019", "13.0687" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "$\\| v \\|=5$ $\\| w \\|=1$ The angle between $v$ and $w$ is 0.3 radians. Given this information, calculate the following:\n(a) $v \\cdotp w$=[ANS]\n(b) $\\| 2 v+4 w \\|=$ [ANS]\n(c) $\\| 2 v-4 w \\|=$ [ANS]", "answer_v2": [ "4.77668", "13.8718", "6.29071" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "$\\| v \\|=4$ $\\| w \\|=2$ The angle between $v$ and $w$ is 1 radians. Given this information, calculate the following:\n(a) $v \\cdotp w$=[ANS]\n(b) $\\| 3 v+1 w \\|=$ [ANS]\n(c) $\\| 2 v-1 w \\|=$ [ANS]", "answer_v3": [ "4.32242", "13.1884", "7.12112" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0048", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "Consider the three points: $A=\\left(8,6\\right)$ $B=\\left(7,8\\right)$ $C=\\left(4,4\\right)$. Determine the angle between $\\overline{AB}$ and $\\overline{AC}$. $\\theta_a$=[ANS]", "answer_v1": [ "1.5708" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the three points: $A=\\left(2,4\\right)$ $B=\\left(6,1\\right)$ $C=\\left(7,5\\right)$. Determine the angle between $\\overline{AB}$ and $\\overline{AC}$. $\\theta_a$=[ANS]", "answer_v2": [ "0.840897" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the three points: $A=\\left(1,6\\right)$ $B=\\left(10,8\\right)$ $C=\\left(7,2\\right)$. Determine the angle between $\\overline{AB}$ and $\\overline{AC}$. $\\theta_a$=[ANS]", "answer_v3": [ "0.806672" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0049", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "4", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "Calculate the force (in Newtons) required to push a 30kg wagon up a 0.6 radian inclined plane. One Newton (N) is equal to 1 $\\frac{kg \\cdotp m}{s^2}$, and the force due to gravity on the wagon is $F=m*g$, where $m$ is the mass of the wagon, and $g$ is the acceleration due to gravity (9.8 $\\frac{m}{s^2}$). Please ignore friction in this problem. Hint: draw a picture, and express the forces on the wagon as vectors. Force=[ANS] $N$", "answer_v1": [ "166.005" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the force (in Newtons) required to push a 50kg wagon up a 0.15 radian inclined plane. One Newton (N) is equal to 1 $\\frac{kg \\cdotp m}{s^2}$, and the force due to gravity on the wagon is $F=m*g$, where $m$ is the mass of the wagon, and $g$ is the acceleration due to gravity (9.8 $\\frac{m}{s^2}$). Please ignore friction in this problem. Hint: draw a picture, and express the forces on the wagon as vectors. Force=[ANS] $N$", "answer_v2": [ "73.2247" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the force (in Newtons) required to push a 40kg wagon up a 0.3 radian inclined plane. One Newton (N) is equal to 1 $\\frac{kg \\cdotp m}{s^2}$, and the force due to gravity on the wagon is $F=m*g$, where $m$ is the mass of the wagon, and $g$ is the acceleration due to gravity (9.8 $\\frac{m}{s^2}$). Please ignore friction in this problem. Hint: draw a picture, and express the forces on the wagon as vectors. Force=[ANS] $N$", "answer_v3": [ "115.844" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0050", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "Assume that $u \\cdotp v=8$, $\\| u \\|=6$, and $\\| v \\|=7$. What is the value of $8 u \\cdotp (4 u-7 v)$? [ANS]", "answer_v1": [ "704" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Assume that $u \\cdotp v=1$, $\\| u \\|=10$, and $\\| v \\|=2$. What is the value of $4 u \\cdotp (10 u-7 v)$? [ANS]", "answer_v2": [ "3972" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Assume that $u \\cdotp v=4$, $\\| u \\|=7$, and $\\| v \\|=3$. What is the value of $6 u \\cdotp (3 u-7 v)$? [ANS]", "answer_v3": [ "714" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0051", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "Compute the dot product $\\left<5,2,3\\right> \\cdotp \\left<5,-4,-4\\right>$. [ANS]", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the dot product $\\left<-9,9,-7\\right> \\cdotp \\left<-3,9,-4\\right>$. [ANS]", "answer_v2": [ "136" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the dot product $\\left<-4,2,-5\\right> \\cdotp \\left<1,-6,-3\\right>$. [ANS]", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0052", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "$v=\\left<8,6,7\\right>$ $w=\\left<8,4,4\\right>$ Find the cosine of the angle between $v$ and $w$. $\\cos{\\theta}$=[ANS]", "answer_v1": [ "0.969905" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$v=\\left<1,10,2\\right>$ $w=\\left<4,10,4\\right>$ Find the cosine of the angle between $v$ and $w$. $\\cos{\\theta}$=[ANS]", "answer_v2": [ "0.951341" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$v=\\left<4,7,3\\right>$ $w=\\left<6,3,4\\right>$ Find the cosine of the angle between $v$ and $w$. $\\cos{\\theta}$=[ANS]", "answer_v3": [ "0.848387" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0053", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "calculus", "parametric", "vector", "dot product", "scalar product", "angle", "projection", "proj" ], "problem_v1": "$v=5\\boldsymbol{i}+2\\boldsymbol{j}+3\\boldsymbol{k}$ $w=5\\boldsymbol{i}-4\\boldsymbol{j}-4\\boldsymbol{k}$ Compute the dot product $v \\cdotp w$. [ANS]", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$v=-9\\boldsymbol{i}+9\\boldsymbol{j}-7\\boldsymbol{k}$ $w=-3\\boldsymbol{i}+9\\boldsymbol{j}-4\\boldsymbol{k}$ Compute the dot product $v \\cdotp w$. [ANS]", "answer_v2": [ "136" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$v=-4\\boldsymbol{i}+2\\boldsymbol{j}-5\\boldsymbol{k}$ $w=\\boldsymbol{i}-6\\boldsymbol{j}-3\\boldsymbol{k}$ Compute the dot product $v \\cdotp w$. [ANS]", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0054", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "$P=\\left(8,6,7\\right)$ $Q=\\left(8,4,4\\right)$ Find $M$, the midpoint of $\\overline{PQ}$. $M$=[ANS]", "answer_v1": [ "(8,5,5.5)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "$P=\\left(1,10,2\\right)$ $Q=\\left(4,10,4\\right)$ Find $M$, the midpoint of $\\overline{PQ}$. $M$=[ANS]", "answer_v2": [ "(2.5,10,3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "$P=\\left(4,7,3\\right)$ $Q=\\left(6,3,4\\right)$ Find $M$, the midpoint of $\\overline{PQ}$. $M$=[ANS]", "answer_v3": [ "(5,5,3.5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0055", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Find a unit vector $\\mathbf{u}$ in the direction opposite of $\\left<-8,-6,-7\\right>$. $\\mathbf{u}$=[ANS]", "answer_v1": [ "(0.655386,0.491539,0.573462)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a unit vector $\\mathbf{u}$ in the direction opposite of $\\left<-1,-10,-2\\right>$. $\\mathbf{u}$=[ANS]", "answer_v2": [ "(0.09759,0.9759,0.19518)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a unit vector $\\mathbf{u}$ in the direction opposite of $\\left<-4,-7,-3\\right>$. $\\mathbf{u}$=[ANS]", "answer_v3": [ "(0.464991,0.813733,0.348743)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0056", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Find the length of the vector $\\mathbf{v}=\\left<8,6,7\\right>$. $\\| \\mathbf{v} \\|$=[ANS]", "answer_v1": [ "12.2066" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the vector $\\mathbf{v}=\\left<1,10,2\\right>$. $\\| \\mathbf{v} \\|$=[ANS]", "answer_v2": [ "10.247" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the vector $\\mathbf{v}=\\left<4,7,3\\right>$. $\\| \\mathbf{v} \\|$=[ANS]", "answer_v3": [ "8.60233" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0057", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "$A=\\left(5,2,3\\right)$ $B=\\left(8,4,4\\right)$ $C=\\left(6,6,4\\right)$ What is the vector $\\mathbf{v}$ whose tail and head are the midpoint of $\\overline{AB}$ and the midpoint of $\\overline{BC}$, respectively. $\\mathbf{v}$=[ANS]", "answer_v1": [ "(0.5,2,0.5)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "$A=\\left(-9,9,-7\\right)$ $B=\\left(4,10,4\\right)$ $C=\\left(2,4,6\\right)$ What is the vector $\\mathbf{v}$ whose tail and head are the midpoint of $\\overline{AB}$ and the midpoint of $\\overline{BC}$, respectively. $\\mathbf{v}$=[ANS]", "answer_v2": [ "(5.5,-2.5,6.5)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "$A=\\left(-4,2,-5\\right)$ $B=\\left(6,3,4\\right)$ $C=\\left(9,10,9\\right)$ What is the vector $\\mathbf{v}$ whose tail and head are the midpoint of $\\overline{AB}$ and the midpoint of $\\overline{BC}$, respectively. $\\mathbf{v}$=[ANS]", "answer_v3": [ "(6.5,4,7)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0058", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Parallel", "Perpendicular", "Plane" ], "problem_v1": "Find a vector $\\overline{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\overline{v}=\\left<3,1,1\\right>$.\n$\\overline{u}$=[ANS].", "answer_v1": [ "(0,1,-1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector $\\overline{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\overline{v}=\\left<-5,5,-4\\right>$.\n$\\overline{u}$=[ANS].", "answer_v2": [ "(0,-4,-5)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector $\\overline{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\overline{v}=\\left<-2,1,-2\\right>$.\n$\\overline{u}$=[ANS].", "answer_v3": [ "(0,-2,-1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0059", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Length" ], "problem_v1": "Find vectors that satisfy the given conditions:\nThe vector in the opposite direction to $\\overline{u}=\\left<3,1\\right>$ and of half its length is [ANS].\nThe vector of length $4$ and in the same direction as $\\overline{v}=\\left<1,2,-2\\right>$ is [ANS].", "answer_v1": [ "(-1.5,-0.5)", "(1.33333,2.66667,-2.66667)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Find vectors that satisfy the given conditions:\nThe vector in the opposite direction to $\\overline{u}=\\left<-5,5\\right>$ and of half its length is [ANS].\nThe vector of length $4$ and in the same direction as $\\overline{v}=\\left<-4,-2,5\\right>$ is [ANS].", "answer_v2": [ "(2.5,-2.5)", "(-2.38514,-1.19257,2.98142)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find vectors that satisfy the given conditions:\nThe vector in the opposite direction to $\\overline{u}=\\left<-2,1\\right>$ and of half its length is [ANS].\nThe vector of length $5$ and in the same direction as $\\overline{v}=\\left<-2,1,-3\\right>$ is [ANS].", "answer_v3": [ "(1,-0.5)", "(-2.67261,1.33631,-4.00892)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0060", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Parallelogram" ], "problem_v1": "Suppose $\\overline{u}=\\left<2,1\\right>$ and $\\overline{v}=\\left<4,-3\\right>$ are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are [ANS]. Separate answers with a comma. Separate answers with a comma.", "answer_v1": [ "(4.47214, 6.32456)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Suppose $\\overline{u}=\\left<-3,3\\right>$ and $\\overline{v}=\\left<3,-9\\right>$ are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are [ANS]. Separate answers with a comma. Separate answers with a comma.", "answer_v2": [ "(13.4164, 6)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Suppose $\\overline{u}=\\left<-1,1\\right>$ and $\\overline{v}=\\left<0,-4\\right>$ are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are [ANS]. Separate answers with a comma. Separate answers with a comma.", "answer_v3": [ "(5.09902, 3.16228)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Geometry_0061", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Unit", "Parallel", "Perpendicular" ], "problem_v1": "In each part, find the two unit vectors in ${\\bf R}^2$ that satisfy the given conditions.\nThe two unit vectors parallel to the line $y=3x+1$ are [ANS] and [ANS].\nThe two unit vectors parallel to the line $3y-2x=1$ are [ANS] and [ANS].\nThe two unit vectors perpendicular to the line $y=x+2$ are [ANS] and [ANS].", "answer_v1": [ "(0.316228,0.948683)", "(-0.316228,-0.948683)", "(-0.83205,-0.5547)", "(0.83205,0.5547)", "(-0.707107,0.707107)", "(0.707107,-0.707107)" ], "answer_type_v1": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "In each part, find the two unit vectors in ${\\bf R}^2$ that satisfy the given conditions.\nThe two unit vectors parallel to the line $y=5-5x$ are [ANS] and [ANS].\nThe two unit vectors parallel to the line $5x+3y=1$ are [ANS] and [ANS].\nThe two unit vectors perpendicular to the line $y=-\\left(4x+2\\right)$ are [ANS] and [ANS].", "answer_v2": [ "(0.196116,-0.980581)", "(-0.196116,0.980581)", "(-0.514496,0.857493)", "(0.514496,-0.857493)", "(0.970143,0.242536)", "(-0.970143,-0.242536)" ], "answer_type_v2": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "In each part, find the two unit vectors in ${\\bf R}^2$ that satisfy the given conditions.\nThe two unit vectors parallel to the line $y=1-2x$ are [ANS] and [ANS].\nThe two unit vectors parallel to the line $3y-3x=1$ are [ANS] and [ANS].\nThe two unit vectors perpendicular to the line $y=1-2x$ are [ANS] and [ANS].", "answer_v3": [ "(0.447214,-0.894427)", "(-0.447214,0.894427)", "(-0.707107,-0.707107)", "(0.707107,0.707107)", "(0.894427,0.447214)", "(-0.894427,-0.447214)" ], "answer_type_v3": [ "OL", "OL", "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Geometry_0062", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Length", "Unit" ], "problem_v1": "Find unit vectors that satisfy the given conditions:\nThe unit vector in the same direction as $\\left<3,1\\right>$ is [ANS].\nThe unit vector oppositely directed to $\\boldsymbol{i}+2\\boldsymbol{j}-2\\boldsymbol{k}$ is [ANS].\nThe unit vector that has the same direction as the vector from the point $A=\\left(-2,1\\right)$ to the point $B=\\left(-1,0\\right)$ is [ANS].", "answer_v1": [ "(0.948683,0.316228)", "(-0.333333,-0.666667,0.666667)", "(0.707107,-0.707107)" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Find unit vectors that satisfy the given conditions:\nThe unit vector in the same direction as $\\left<-5,5\\right>$ is [ANS].\nThe unit vector oppositely directed to $-4\\boldsymbol{i}-2\\boldsymbol{j}+5\\boldsymbol{k}$ is [ANS].\nThe unit vector that has the same direction as the vector from the point $A=\\left(-2,-3\\right)$ to the point $B=\\left(-4,-2\\right)$ is [ANS].", "answer_v2": [ "(-0.707107,0.707107)", "(0.596285,0.298142,-0.745356)", "(-0.894427,0.447214)" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Find unit vectors that satisfy the given conditions:\nThe unit vector in the same direction as $\\left<-2,1\\right>$ is [ANS].\nThe unit vector oppositely directed to $-2\\boldsymbol{i}+\\boldsymbol{j}-3\\boldsymbol{k}$ is [ANS].\nThe unit vector that has the same direction as the vector from the point $A=\\left(-2,3\\right)$ to the point $B=\\left(3,7\\right)$ is [ANS].", "answer_v3": [ "(-0.894427,0.447214)", "(0.534522,-0.267261,0.801784)", "(0.780869,0.624695)" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0063", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Perpendicular", "Dot Product" ], "problem_v1": "Suppose $\\overline{u}=\\left<4,2,1\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\overline{u}$, and an \" F \" if it is not perpendicular to $\\overline{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<2,-2,-2\\right> \\\\\\hline [ANS] & 2. & \\left<1,2,-8\\right> \\\\\\hline [ANS] & 3. & \\left<1,-2,0\\right> \\\\\\hline [ANS] & 4. & \\left<1,1,-1\\right> \\\\\\hline\\end{array}$", "answer_v1": [ "F", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $\\overline{u}=\\left<-6,6,-4\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\overline{u}$, and an \" F \" if it is not perpendicular to $\\overline{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<3,3,0\\right> \\\\\\hline [ANS] & 2. & \\left<-2,5,-2\\right> \\\\\\hline [ANS] & 3. & \\left<10,-18,-42\\right> \\\\\\hline [ANS] & 4. & \\left<-3,-2,1\\right> \\\\\\hline\\end{array}$", "answer_v2": [ "T", "F", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $\\overline{u}=\\left<-2,2,-2\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\overline{u}$, and an \" F \" if it is not perpendicular to $\\overline{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<1,-3,-2\\right> \\\\\\hline [ANS] & 2. & \\left<4,-10,-14\\right> \\\\\\hline [ANS] & 3. & \\left<3,5,4\\right> \\\\\\hline [ANS] & 4. & \\left<1,1,0\\right> \\\\\\hline\\end{array}$", "answer_v3": [ "F", "T", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0064", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "Vector", "Tip", "Tail", "Length" ], "problem_v1": "Consider the vector $\\overline{v}$ between $\\left(2.6,0.8\\right)$ and $\\left(1.2,2.2\\right)$.\nThe vector $\\overline{v}$ is [ANS].\nThe length of $\\overline{v}$ is [ANS].\nIf the tail of $\\overline{v}$ is at $\\left(-2,-1.7\\right)$, then the tip is at [ANS].\nIf the tip of $\\overline{v}$ is at $\\left(0.7,0.8\\right)$ then its tail is at [ANS].\nWhat vector has the same length as $\\overline{v}$, but points in the opposite direction? [ANS].\nWhat vector has the same direction as $\\overline{v}$, but is twice as long? [ANS].", "answer_v1": [ "(-1.4,1.4)", "1.9799", "(-3.4,-0.3)", "(2.1,-0.6)", "(1.4,-1.4)", "(-2.8,2.8)" ], "answer_type_v1": [ "OL", "NV", "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consider the vector $\\overline{v}$ between $\\left(-4.2,4.4\\right)$ and $\\left(-3.5,-1.7\\right)$.\nThe vector $\\overline{v}$ is [ANS].\nThe length of $\\overline{v}$ is [ANS].\nIf the tail of $\\overline{v}$ is at $\\left(4.5,-1.9\\right)$, then the tip is at [ANS].\nIf the tip of $\\overline{v}$ is at $\\left(-3.2,-1.7\\right)$ then its tail is at [ANS].\nWhat vector has the same length as $\\overline{v}$, but points in the opposite direction? [ANS].\nWhat vector has the same direction as $\\overline{v}$, but is twice as long? [ANS].", "answer_v2": [ "(0.7,-6.1)", "6.14003", "(5.2,-8)", "(-3.9,4.4)", "(-0.7,6.1)", "(1.4,-12.2)" ], "answer_type_v2": [ "OL", "NV", "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consider the vector $\\overline{v}$ between $\\left(-1.9,1.1\\right)$ and $\\left(-2.2,0.5\\right)$.\nThe vector $\\overline{v}$ is [ANS].\nThe length of $\\overline{v}$ is [ANS].\nIf the tail of $\\overline{v}$ is at $\\left(-3,-1.6\\right)$, then the tip is at [ANS].\nIf the tip of $\\overline{v}$ is at $\\left(3.1,4.2\\right)$ then its tail is at [ANS].\nWhat vector has the same length as $\\overline{v}$, but points in the opposite direction? [ANS].\nWhat vector has the same direction as $\\overline{v}$, but is twice as long? [ANS].", "answer_v3": [ "(-0.3,-0.6)", "0.67082", "(-3.3,-2.2)", "(3.4,4.8)", "(0.3,0.6)", "(-0.6,-1.2)" ], "answer_type_v3": [ "OL", "NV", "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Geometry_0065", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Vector", "Projection", "Orthogonal" ], "problem_v1": "Suppose $\\overline{u}=\\left<3,1,1\\right>$ and $\\overline{v}=\\left<2,-2,-2\\right>$. Then:\nThe projection of $\\overline{u}$ along $\\overline{v}$ is [ANS].\nThe projection of $\\overline{u}$ orthogonal to $\\overline{v}$ is [ANS].", "answer_v1": [ "(0.333333,-0.333333,-0.333333)", "(2.66667,1.33333,1.33333)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $\\overline{u}=\\left<-5,5,-4\\right>$ and $\\overline{v}=\\left<-2,5,-2\\right>$. Then:\nThe projection of $\\overline{u}$ along $\\overline{v}$ is [ANS].\nThe projection of $\\overline{u}$ orthogonal to $\\overline{v}$ is [ANS].", "answer_v2": [ "(-2.60606,6.51515,-2.60606)", "(-2.39394,-1.51515,-1.39394)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $\\overline{u}=\\left<-2,1,-2\\right>$ and $\\overline{v}=\\left<1,-3,-2\\right>$. Then:\nThe projection of $\\overline{u}$ along $\\overline{v}$ is [ANS].\nThe projection of $\\overline{u}$ orthogonal to $\\overline{v}$ is [ANS].", "answer_v3": [ "(-0.0714286,0.214286,0.142857)", "(-1.92857,0.785714,-2.14286)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0066", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "5", "keywords": [ "Vector", "Speed", "Direction", "Introduction" ], "problem_v1": "The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 65 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)?\nspeed=[ANS]", "answer_v1": [ "91.9238815542512" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 51 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)?\nspeed=[ANS]", "answer_v2": [ "72.1248916810278" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The nine Ring Wraiths want to fly from Barad-Dur to Rivendell. Rivendell is directly north of Barad-Dur. The Dark Tower reports that the wind is coming from the west at 56 miles per hour. In order to travel in a straight line, the Ring Wraiths decide to head northwest. At what speed should they fly (omit units)?\nspeed=[ANS]", "answer_v3": [ "79.1959594928933" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0067", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Dot Product", "Force", "Work", "vector" ], "problem_v1": "A woman exerts a horizontal force of 8 pounds on a box as she pushes it up a ramp that is 6 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. Work: [ANS] ft-lb", "answer_v1": [ "41.5692193816531" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A woman exerts a horizontal force of 1 pounds on a box as she pushes it up a ramp that is 10 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. Work: [ANS] ft-lb", "answer_v2": [ "8.66025403784439" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A woman exerts a horizontal force of 4 pounds on a box as she pushes it up a ramp that is 7 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. Work: [ANS] ft-lb", "answer_v3": [ "24.2487113059643" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0068", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Dot Product", "Projection", "Scalar", "Vector", "Orthogonal", "vector" ], "problem_v1": "Let ${\\mathbf a}=(5, 2, 3)$ and ${\\mathbf b}=(5,-4,-4)$ be vectors. (A) Find the scalar projection of ${\\mathbf b}$ onto ${\\mathbf a}$.\nScalar Projection: [ANS]\n(B) Decompose the vector ${\\mathbf b}$ into a component parallel to ${\\mathbf a}$ and a component orthogonal to ${\\mathbf a}$.\nParallel component: ([ANS], [ANS], [ANS]) Orthogonal Component: ([ANS], [ANS], [ANS])", "answer_v1": [ "0.811107105653813", "0.657894736842105", "0.263157894736842", "0.394736842105263", "4.34210526315789", "-4.26315789473684", "-4.39473684210526" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Let ${\\mathbf a}=(-9, 9,-7)$ and ${\\mathbf b}=(-3, 9,-4)$ be vectors. (A) Find the scalar projection of ${\\mathbf b}$ onto ${\\mathbf a}$.\nScalar Projection: [ANS]\n(B) Decompose the vector ${\\mathbf b}$ into a component parallel to ${\\mathbf a}$ and a component orthogonal to ${\\mathbf a}$.\nParallel component: ([ANS], [ANS], [ANS]) Orthogonal Component: ([ANS], [ANS], [ANS])", "answer_v2": [ "9.36262611517259", "-5.80094786729858", "5.80094786729858", "-4.51184834123223", "2.80094786729858", "3.19905213270142", "0.511848341232228" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Let ${\\mathbf a}=(-4, 2,-5)$ and ${\\mathbf b}=(1,-6,-3)$ be vectors. (A) Find the scalar projection of ${\\mathbf b}$ onto ${\\mathbf a}$.\nScalar Projection: [ANS]\n(B) Decompose the vector ${\\mathbf b}$ into a component parallel to ${\\mathbf a}$ and a component orthogonal to ${\\mathbf a}$.\nParallel component: ([ANS], [ANS], [ANS]) Orthogonal Component: ([ANS], [ANS], [ANS])", "answer_v3": [ "-0.149071198499986", "0.0888888888888889", "-0.0444444444444444", "0.111111111111111", "0.911111111111111", "-5.95555555555556", "-3.11111111111111" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Geometry_0069", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Scale", "Unit", "Vector", "vector" ], "problem_v1": "Find a unit vector in the same direction as ${\\mathbf a}=(5, 2, 3)$.\n([ANS], [ANS], [ANS])", "answer_v1": [ "0.811107105653813", "0.324442842261525", "0.486664263392288" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a unit vector in the same direction as ${\\mathbf a}=(-9, 9,-7)$.\n([ANS], [ANS], [ANS])", "answer_v2": [ "-0.619585551739363", "0.619585551739363", "-0.48189987357506" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a unit vector in the same direction as ${\\mathbf a}=(-4, 2,-5)$.\n([ANS], [ANS], [ANS])", "answer_v3": [ "-0.596284793999944", "0.298142396999972", "-0.74535599249993" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0070", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "Dot Product", "vector" ], "problem_v1": "If ${\\mathbf a}$=(5, 2, 3) and ${\\mathbf b}$=(5,-4,-4), find ${\\mathbf a \\cdot \\mathbf b}$=[ANS].", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If ${\\mathbf a}$=(-9, 9,-7) and ${\\mathbf b}$=(-3, 9,-4), find ${\\mathbf a \\cdot \\mathbf b}$=[ANS].", "answer_v2": [ "136" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If ${\\mathbf a}$=(-4, 2,-5) and ${\\mathbf b}$=(1,-6,-3), find ${\\mathbf a \\cdot \\mathbf b}$=[ANS].", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0071", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "Dot Product", "Angle", "Norm", "Length" ], "problem_v1": "Find ${\\mathbf a \\cdot \\mathbf b}$ if $\\left| {\\mathbf a} \\right|$=8, $\\left| {\\mathbf b} \\right|$=6, and the angle between ${\\mathbf a}$ and ${\\mathbf b}$ is $\\frac{\\pi}{3}$ radians.\n${\\mathbf a \\cdot \\mathbf b}$=[ANS]", "answer_v1": [ "24" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find ${\\mathbf a \\cdot \\mathbf b}$ if $\\left| {\\mathbf a} \\right|$=1, $\\left| {\\mathbf b} \\right|$=10, and the angle between ${\\mathbf a}$ and ${\\mathbf b}$ is $-\\frac{\\pi}{7}$ radians.\n${\\mathbf a \\cdot \\mathbf b}$=[ANS]", "answer_v2": [ "9.00968867902419" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find ${\\mathbf a \\cdot \\mathbf b}$ if $\\left| {\\mathbf a} \\right|$=4, $\\left| {\\mathbf b} \\right|$=7, and the angle between ${\\mathbf a}$ and ${\\mathbf b}$ is $-\\frac{\\pi}{5}$ radians.\n${\\mathbf a \\cdot \\mathbf b}$=[ANS]", "answer_v3": [ "22.6524758424985" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0072", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "4", "keywords": [ "Dot Product", "Force", "Work", "vector" ], "problem_v1": "A constant force ${\\mathbf F}=6 {\\mathbf i}+3 {\\mathbf j}+3 {\\mathbf k}$ moves an object along a straight line from point $(4,-3,-3)$ to point $(5, 0,-7)$. Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.\nWork: [ANS] Nm", "answer_v1": [ "3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A constant force ${\\mathbf F}=-8 {\\mathbf i}+8 {\\mathbf j}-7 {\\mathbf k}$ moves an object along a straight line from point $(-3, 8,-3)$ to point $(-5, 2, 5)$. Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.\nWork: [ANS] Nm", "answer_v2": [ "-88" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A constant force ${\\mathbf F}=-4 {\\mathbf i}+3 {\\mathbf j}-5 {\\mathbf k}$ moves an object along a straight line from point $(1,-5,-3)$ to point $(3, 7,-7)$. Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.\nWork: [ANS] Nm", "answer_v3": [ "48" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0073", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "Dot Product", "Angle", "\"vector" ], "problem_v1": "What is the angle in radians between the vectors ${\\mathbf a}$=(5, 2, 3) and ${\\mathbf b}$=(5,-4,-4)? Angle: [ANS] (radians)", "answer_v1": [ "1.463154818329" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the angle in radians between the vectors ${\\mathbf a}$=(-9, 9,-7) and ${\\mathbf b}$=(-3, 9,-4)? Angle: [ANS] (radians)", "answer_v2": [ "0.429008481243571" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the angle in radians between the vectors ${\\mathbf a}$=(-4, 2,-5) and ${\\mathbf b}$=(1,-6,-3)? Angle: [ANS] (radians)", "answer_v3": [ "1.59277744596675" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0074", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "5", "keywords": [ "Dot Product", "Angle", "vector" ], "problem_v1": "A child walks due east on the deck of a ship at 4 miles per hour. The ship is moving north at a speed of 12 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed=[ANS] mph The angle of the direction from the north=[ANS] (radians)", "answer_v1": [ "12.6491106406735", "0.321750554396642" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A child walks due east on the deck of a ship at 1 miles per hour. The ship is moving north at a speed of 19 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed=[ANS] mph The angle of the direction from the north=[ANS] (radians)", "answer_v2": [ "19.0262975904404", "0.0525830616109411" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A child walks due east on the deck of a ship at 2 miles per hour. The ship is moving north at a speed of 13 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed=[ANS] mph The angle of the direction from the north=[ANS] (radians)", "answer_v3": [ "13.1529464379659", "0.152649328395265" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0075", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "4", "keywords": [ "Dot Product", "Tension", "Force", "Mass", "vector" ], "problem_v1": "A horizontal clothesline is tied between 2 poles, 16 meters apart. When a mass of 4 kilograms is tied to the middle of the clothesline, it sags a distance of 4 meters. What is the magnitude of the tension on the ends of the clothesline? Tension=[ANS] N", "answer_v1": [ "43.8269323589959" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A horizontal clothesline is tied between 2 poles, 20 meters apart. When a mass of 1 kilograms is tied to the middle of the clothesline, it sags a distance of 1 meters. What is the magnitude of the tension on the ends of the clothesline? Tension=[ANS] N", "answer_v2": [ "49.2443905434924" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A horizontal clothesline is tied between 2 poles, 16 meters apart. When a mass of 2 kilograms is tied to the middle of the clothesline, it sags a distance of 2 meters. What is the magnitude of the tension on the ends of the clothesline? Tension=[ANS] N", "answer_v3": [ "40.4064351310531" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0076", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "Dot Product", "Angle" ], "problem_v1": "A vertical force of 8 (pounds) moves an object from point (7,5,5) to (11,-1,8), where distance is measured in feet.\nThe force vector is F=[ANS]\nThe displacement vector is d=[ANS]\nThe work done by the force on the object is F $\\cdot$ d=[ANS] ftlbs", "answer_v1": [ "(0,0,8)", "(4,-6,3)", "24" ], "answer_type_v1": [ "OL", "OL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A vertical force of 6 (pounds) moves an object from point (1,8,2) to (5,7,6), where distance is measured in feet.\nThe force vector is F=[ANS]\nThe displacement vector is d=[ANS]\nThe work done by the force on the object is F $\\cdot$ d=[ANS] ftlbs", "answer_v2": [ "(0,0,6)", "(4,-1,4)", "24" ], "answer_type_v2": [ "OL", "OL", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A vertical force of 9 (pounds) moves an object from point (3,5,3) to (8,-2,7), where distance is measured in feet.\nThe force vector is F=[ANS]\nThe displacement vector is d=[ANS]\nThe work done by the force on the object is F $\\cdot$ d=[ANS] ftlbs", "answer_v3": [ "(0,0,9)", "(5,-7,4)", "36" ], "answer_type_v3": [ "OL", "OL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0077", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "Find the length of the vectors\n(a) $3\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$: length=[ANS]\n(b) $1.4\\,\\mathit{\\vec i}-1.2\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k}$: length=[ANS]", "answer_v1": [ "3.31662", "2.09762" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the length of the vectors\n(a) $-5\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}-4\\,\\mathit{\\vec k}$: length=[ANS]\n(b) $-\\,\\mathit{\\vec i}+2.8\\,\\mathit{\\vec j}-1.2\\,\\mathit{\\vec k}$: length=[ANS]", "answer_v2": [ "8.12404", "3.20624" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the length of the vectors\n(a) $-2\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$: length=[ANS]\n(b) $0.4\\,\\mathit{\\vec i}-1.8\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k}$: length=[ANS]", "answer_v3": [ "3", "2.09762" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0078", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "(a) Find a unit vector from the point $P=(3,2)$ and toward the point $Q=(15,7)$. $\\vec u=$ [ANS]\n(b) Find a vector of length 39 pointing in the same direction. $\\vec v=$ [ANS]", "answer_v1": [ "0.923077i+0.384615j", "36i+15j" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find a unit vector from the point $P=(1,3)$ and toward the point $Q=(4,7)$. $\\vec u=$ [ANS]\n(b) Find a vector of length 10 pointing in the same direction. $\\vec v=$ [ANS]", "answer_v2": [ "0.6i+0.8j", "6i+8j" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find a unit vector from the point $P=(1,2)$ and toward the point $Q=(5,5)$. $\\vec u=$ [ANS]\n(b) Find a vector of length 15 pointing in the same direction. $\\vec v=$ [ANS]", "answer_v3": [ "0.8i+0.6j", "12i+9j" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0079", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors", "coordinate vector" ], "problem_v1": "Find all vectors $\\vec v$ in 2 dimensions having $||\\vec v||=41$ where the $\\,\\mathit{\\vec j}$-component of $\\vec v$ is $9 \\,\\mathit{\\vec j}$. vectors: [ANS]\n(If you find more than one vector, enter them in a comma-separated list.) (If you find more than one vector, enter them in a comma-separated list.)", "answer_v1": [ "(40i+9j, -40i+9j)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find all vectors $\\vec v$ in 2 dimensions having $||\\vec v||=5$ where the $\\,\\mathit{\\vec j}$-component of $\\vec v$ is $3 \\,\\mathit{\\vec j}$. vectors: [ANS]\n(If you find more than one vector, enter them in a comma-separated list.) (If you find more than one vector, enter them in a comma-separated list.)", "answer_v2": [ "(4i+3j, -4i+3j)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find all vectors $\\vec v$ in 2 dimensions having $||\\vec v||=13$ where the $\\,\\mathit{\\vec j}$-component of $\\vec v$ is $5 \\,\\mathit{\\vec j}$. vectors: [ANS]\n(If you find more than one vector, enter them in a comma-separated list.) (If you find more than one vector, enter them in a comma-separated list.)", "answer_v3": [ "(12i+5j, -12i+5j)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0080", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "5", "keywords": [ "vectors", "force" ], "problem_v1": "A plane is heading due south and climbing at the rate of 110 km/hr. If its airspeed is 510 km/hr and there is a wind blowing 110 km/hr to the northwest, what is the ground speed of the plane? ground speed=[ANS]", "answer_v1": [ "427.352" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A plane is heading due east and climbing at the rate of 50 km/hr. If its airspeed is 550 km/hr and there is a wind blowing 80 km/hr to the northwest, what is the ground speed of the plane? ground speed=[ANS]", "answer_v2": [ "494.401" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A plane is heading due south and climbing at the rate of 70 km/hr. If its airspeed is 510 km/hr and there is a wind blowing 90 km/hr to the northwest, what is the ground speed of the plane? ground speed=[ANS]", "answer_v3": [ "446.096" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0081", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "5", "keywords": [ "vectors", "force" ], "problem_v1": "A man wishes to row the shortest possible distance from north to south across a river which is flowing at 5 km/hr from the east. He can row at 7 km/hr.\n(a) In which direction should he steer? At an angle (in degrees) of [ANS] [ANS] of south. (b) Suppose that when the man is in the middle of the river, a wind of 12 km/hr from the southwest starts. In which direction should he steer to move straight across the river? At an angle (in degrees) of [ANS] [ANS] of south. Which bank of the river will the rower reach first? [ANS] bank.", "answer_v1": [ "180*asin(5/7)/pi", "east", "180*asin((12/[sqrt(2)]-5)/7)/pi", "west", "The north" ], "answer_type_v1": [ "NV", "MCS", "NV", "MCS", "MCS" ], "options_v1": [ [], [ "west", "east" ], [], [ "west", "east" ], [ "The north", "The south", "Neither" ] ], "problem_v2": "A man wishes to row the shortest possible distance from north to south across a river which is flowing at 2 km/hr from the east. He can row at 4 km/hr.\n(a) In which direction should he steer? At an angle (in degrees) of [ANS] [ANS] of south. (b) Suppose that when the man is in the middle of the river, a wind of 5 km/hr from the southwest starts. In which direction should he steer to move straight across the river? At an angle (in degrees) of [ANS] [ANS] of south. Which bank of the river will the rower reach first? [ANS] bank.", "answer_v2": [ "180*asin(2/4)/pi", "east", "180*asin((5/[sqrt(2)]-2)/4)/pi", "west", "The south" ], "answer_type_v2": [ "NV", "MCS", "NV", "MCS", "MCS" ], "options_v2": [ [], [ "west", "east" ], [], [ "west", "east" ], [ "The north", "The south", "Neither" ] ], "problem_v3": "A man wishes to row the shortest possible distance from north to south across a river which is flowing at 3 km/hr from the east. He can row at 5 km/hr.\n(a) In which direction should he steer? At an angle (in degrees) of [ANS] [ANS] of south. (b) Suppose that when the man is in the middle of the river, a wind of 6 km/hr from the southwest starts. In which direction should he steer to move straight across the river? At an angle (in degrees) of [ANS] [ANS] of south. Which bank of the river will the rower reach first? [ANS] bank.", "answer_v3": [ "180*asin(3/5)/pi", "east", "180*asin((6/[sqrt(2)]-3)/5)/pi", "west", "The south" ], "answer_type_v3": [ "NV", "MCS", "NV", "MCS", "MCS" ], "options_v3": [ [], [ "west", "east" ], [], [ "west", "east" ], [ "The north", "The south", "Neither" ] ] }, { "id": "Geometry_0082", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors", "force" ], "problem_v1": "Which is traveling faster, a car whose velocity vector is $28\\vec i+33\\vec j$, or a car whose velocity vector is $40\\vec i$, assuming that the units are the same for both directions? [ANS] is the faster car. At what speed is the faster car traveling? speed=[ANS]", "answer_v1": [ "the first car", "43.2782" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "the first car", "the second car" ], [] ], "problem_v2": "Which is traveling faster, a car whose velocity vector is $20\\vec i+25\\vec j$, or a car whose velocity vector is $30\\vec i$, assuming that the units are the same for both directions? [ANS] is the faster car. At what speed is the faster car traveling? speed=[ANS]", "answer_v2": [ "the first car", "32.0156" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "the first car", "the second car" ], [] ], "problem_v3": "Which is traveling faster, a car whose velocity vector is $23\\vec i+28\\vec j$, or a car whose velocity vector is $30\\vec i$, assuming that the units are the same for both directions? [ANS] is the faster car. At what speed is the faster car traveling? speed=[ANS]", "answer_v3": [ "the first car", "36.2353" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "the first car", "the second car" ], [] ] }, { "id": "Geometry_0083", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors", "dot product" ], "problem_v1": "Find a vector $\\vec w$ that bisects the smaller of the two angles formed by $15\\,\\vec i+8\\,\\vec j$ and $40\\,\\vec i-9\\,\\vec j$. $\\vec w=$ [ANS]", "answer_v1": [ "1.85796i+0.251076j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a vector $\\vec w$ that bisects the smaller of the two angles formed by $3\\,\\vec i+4\\,\\vec j$ and $9\\,\\vec i-40\\,\\vec j$. $\\vec w=$ [ANS]", "answer_v2": [ "0.819512i-0.17561j" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a vector $\\vec w$ that bisects the smaller of the two angles formed by $3\\,\\vec i+4\\,\\vec j$ and $15\\,\\vec i-8\\,\\vec j$. $\\vec w=$ [ANS]", "answer_v3": [ "1.48235i+0.329412j" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0085", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "4", "keywords": [ "vectors", "dot product" ], "problem_v1": "A basketball gymnasium is 40 meters high, 80 meters wide and 210 meters long. For a half time stunt, the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the angle made by the strings as they cross? $\\cos(\\theta)=$ [ANS]", "answer_v1": [ "(-80^2+210^2+40^2)/(80^2+210^2+40^2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A basketball gymnasium is 25 meters high, 90 meters wide and 160 meters long. For a half time stunt, the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the angle made by the strings as they cross? $\\cos(\\theta)=$ [ANS]", "answer_v2": [ "(-90^2+160^2+25^2)/(90^2+160^2+25^2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A basketball gymnasium is 30 meters high, 80 meters wide and 180 meters long. For a half time stunt, the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the angle made by the strings as they cross? $\\cos(\\theta)=$ [ANS]", "answer_v3": [ "(-80^2+180^2+30^2)/(80^2+180^2+30^2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0086", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors", "dot product" ], "problem_v1": "The force on an object is $\\vec F=-13\\,\\mathit{\\vec j}$. For the vector $\\vec v=\\,\\mathit{\\vec i}-\\,\\mathit{\\vec j}$, find:\n(a) The component of $\\vec F$ parallel to $\\vec v$: [ANS]\n(b) The component of $\\vec F$ perpendicular to $\\vec v$: [ANS]\nThe work, $W$, done by force $\\vec F$ through displacement $\\vec v$: [ANS]", "answer_v1": [ "6.5i-6.5j", "-6.5i-6.5j", "13" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The force on an object is $\\vec F=-24\\,\\mathit{\\vec j}$. For the vector $\\vec v=5\\,\\mathit{\\vec i}-4\\,\\mathit{\\vec j}$, find:\n(a) The component of $\\vec F$ parallel to $\\vec v$: [ANS]\n(b) The component of $\\vec F$ perpendicular to $\\vec v$: [ANS]\nThe work, $W$, done by force $\\vec F$ through displacement $\\vec v$: [ANS]", "answer_v2": [ "11.7073i-9.36585j", "-11.7073i-14.6341j", "96" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The force on an object is $\\vec F=-20\\,\\mathit{\\vec j}$. For the vector $\\vec v=\\,\\mathit{\\vec i}-2\\,\\mathit{\\vec j}$, find:\n(a) The component of $\\vec F$ parallel to $\\vec v$: [ANS]\n(b) The component of $\\vec F$ perpendicular to $\\vec v$: [ANS]\nThe work, $W$, done by force $\\vec F$ through displacement $\\vec v$: [ANS]", "answer_v3": [ "8i-16j", "-8i-4j", "40" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0087", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors", "dot product" ], "problem_v1": "Let $\\vec a$, $\\vec b$, $\\vec c$ and $\\vec y$ be the three dimensional vectors \\vec a=4\\,\\mathit{\\vec j}+3\\,\\mathit{\\vec k},\\quad \\vec b=\\,\\mathit{\\vec i}+6\\,\\mathit{\\vec j}+3\\,\\mathit{\\vec k},\\quad \\vec c=2\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j},\\quad \\vec y=\\,\\mathit{\\vec i}-2\\,\\mathit{\\vec j} Perform the following operations on these vectors:\n(a) $\\vec c\\cdot\\vec a+\\vec a\\cdot\\vec y=$ [ANS]\n(b) $(\\vec a\\cdot\\vec b)\\,\\vec a=$ [ANS]\n(c) $((\\vec c\\cdot\\vec c)\\,\\vec a)\\cdot\\vec a=$ [ANS]", "answer_v1": [ "-4", "132j+99k", "125" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\vec a$, $\\vec b$, $\\vec c$ and $\\vec y$ be the three dimensional vectors \\vec a=\\,\\mathit{\\vec j}+5\\,\\mathit{\\vec k},\\quad \\vec b=-4\\,\\mathit{\\vec i}+3\\,\\mathit{\\vec j}+8\\,\\mathit{\\vec k},\\quad \\vec c=2\\,\\mathit{\\vec i}-5\\,\\mathit{\\vec j},\\quad \\vec y=-3\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j} Perform the following operations on these vectors:\n(a) $\\vec c\\cdot\\vec a+\\vec a\\cdot\\vec y=$ [ANS]\n(b) $(\\vec a\\cdot\\vec b)\\,\\vec a=$ [ANS]\n(c) $((\\vec c\\cdot\\vec c)\\,\\vec a)\\cdot\\vec a=$ [ANS]", "answer_v2": [ "-4", "43j+215k", "754" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\vec a$, $\\vec b$, $\\vec c$ and $\\vec y$ be the three dimensional vectors \\vec a=2\\,\\mathit{\\vec j}+4\\,\\mathit{\\vec k},\\quad \\vec b=-2\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}+2\\,\\mathit{\\vec k},\\quad \\vec c=2\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j},\\quad \\vec y=7\\,\\mathit{\\vec i}+6\\,\\mathit{\\vec j} Perform the following operations on these vectors:\n(a) $\\vec c\\cdot\\vec a+\\vec a\\cdot\\vec y=$ [ANS]\n(b) $(\\vec a\\cdot\\vec b)\\,\\vec a=$ [ANS]\n(c) $((\\vec c\\cdot\\vec c)\\,\\vec a)\\cdot\\vec a=$ [ANS]", "answer_v3": [ "22", "36j+72k", "580" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0088", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "5", "keywords": [ "calculus", "vector", "Dot Product", "Projection" ], "problem_v1": "Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates $(2, 1)$ and arrived in the Iron Hills at the point with coordinates $(4, 6)$. If he began walking in the direction of the vector $\\mathbf{v}=3 \\mathbf{i}+1 \\mathbf{j}$ and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. ([ANS], [ANS])", "answer_v1": [ "5.3", "2.1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates $(-3, 3)$ and arrived in the Iron Hills at the point with coordinates $(-2, 7)$. If he began walking in the direction of the vector $\\mathbf{v}=5 \\mathbf{i}+1 \\mathbf{j}$ and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. ([ANS], [ANS])", "answer_v2": [ "-1.26923076923077", "3.34615384615385" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates $(-1, 1)$ and arrived in the Iron Hills at the point with coordinates $(0, 5)$. If he began walking in the direction of the vector $\\mathbf{v}=3 \\mathbf{i}+1 \\mathbf{j}$ and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. ([ANS], [ANS])", "answer_v3": [ "1.1", "1.7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0089", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Find a fully reduced equation for the set of all points in $\\mathbb{R}^2$ (i.e., the $xy$-plane) that are equidistant from the points $A=\\left(3,1\\right)$ and $B=\\left(1,2\\right)$. Recall that a point $P=(x,y)$ is equidistant from $A$ and $B$ if $d(P,A)=d(P,B)$. [ANS]", "answer_v1": [ "2*y-4*x = -5" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find a fully reduced equation for the set of all points in $\\mathbb{R}^2$ (i.e., the $xy$-plane) that are equidistant from the points $A=\\left(-5,5\\right)$ and $B=\\left(-4,-2\\right)$. Recall that a point $P=(x,y)$ is equidistant from $A$ and $B$ if $d(P,A)=d(P,B)$. [ANS]", "answer_v2": [ "2*x-14*y = -30" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find a fully reduced equation for the set of all points in $\\mathbb{R}^2$ (i.e., the $xy$-plane) that are equidistant from the points $A=\\left(-2,1\\right)$ and $B=\\left(-3,-2\\right)$. Recall that a point $P=(x,y)$ is equidistant from $A$ and $B$ if $d(P,A)=d(P,B)$. [ANS]", "answer_v3": [ "2*x+6*y = -8" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Geometry_0090", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors" ], "problem_v1": "Let $A=\\left(1,-2\\right)$ and $B=\\left(2,1\\right)$ be points in $\\mathbb{R}^2$, and let $C=\\left(1,0,-2\\right)$ and $D=\\left(-1,1,0\\right)$ be points in $\\mathbb{R}^3$.\nFind the distance between $A$ and $B$. [ANS]\nFind the distance between $C$ and $D$. [ANS]\nFind the distance between the point $(x,y,z)$ and $C$. [ANS]\nFind the length of the vector $\\overrightarrow{CD}$. [ANS]\nFind the length of the vector $5 \\overrightarrow{BA}$. [ANS]", "answer_v1": [ "sqrt(10)", "sqrt(9)", "sqrt((1-x)^2+(0-y)^2+(-2-z)^2)", "sqrt(9)", "5*sqrt(10)" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $A=\\left(4,-1\\right)$ and $B=\\left(-3,4\\right)$ be points in $\\mathbb{R}^2$, and let $C=\\left(-2,-2,-4\\right)$ and $D=\\left(-3,1,1\\right)$ be points in $\\mathbb{R}^3$.\nFind the distance between $A$ and $B$. [ANS]\nFind the distance between $C$ and $D$. [ANS]\nFind the distance between the point $(x,y,z)$ and $C$. [ANS]\nFind the length of the vector $\\overrightarrow{CD}$. [ANS]\nFind the length of the vector $2 \\overrightarrow{BA}$. [ANS]", "answer_v2": [ "sqrt(74)", "sqrt(35)", "sqrt((-2-x)^2+(-2-y)^2+(-4-z)^2)", "sqrt(35)", "2*sqrt(74)" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $A=\\left(1,0\\right)$ and $B=\\left(-2,-3\\right)$ be points in $\\mathbb{R}^2$, and let $C=\\left(-1,4,-3\\right)$ and $D=\\left(3,3,-2\\right)$ be points in $\\mathbb{R}^3$.\nFind the distance between $A$ and $B$. [ANS]\nFind the distance between $C$ and $D$. [ANS]\nFind the distance between the point $(x,y,z)$ and $C$. [ANS]\nFind the length of the vector $\\overrightarrow{CD}$. [ANS]\nFind the length of the vector $3 \\overrightarrow{BA}$. [ANS]", "answer_v3": [ "sqrt(18)", "sqrt(18)", "sqrt((-1-x)^2+(4-y)^2+(-3-z)^2)", "sqrt(18)", "3*sqrt(18)" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Geometry_0091", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors" ], "problem_v1": "Find the lengths of the three sides of the triangle $ABC$ in $\\mathbb{R}^3$ if $A=\\left(3,1,1\\right)$, $B=\\left(2,-2,-2\\right)$ and $C=\\left(1,1,-1\\right)$. Enter your answers as a comma separated list. [ANS]", "answer_v1": [ "(4.3589, 2.82843, 3.31662)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find the lengths of the three sides of the triangle $ABC$ in $\\mathbb{R}^3$ if $A=\\left(-5,5,-4\\right)$, $B=\\left(-2,5,-2\\right)$ and $C=\\left(-3,-2,1\\right)$. Enter your answers as a comma separated list. [ANS]", "answer_v2": [ "(3.60555, 8.83176, 7.68115)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find the lengths of the three sides of the triangle $ABC$ in $\\mathbb{R}^3$ if $A=\\left(-2,1,-2\\right)$, $B=\\left(1,-3,-2\\right)$ and $C=\\left(3,5,4\\right)$. Enter your answers as a comma separated list. [ANS]", "answer_v3": [ "(5, 8.77496, 10.198)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Geometry_0092", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "inner product' 'vector" ], "problem_v1": "Distance and Dot Products: Consider the vectors {\\mathbf u}=\\langle5, 2, 3\\rangle \\textrm{and} {\\mathbf v}=\\langle 5,-4,-4\\rangle. Compute $\\Vert \\mathbf{u}\\Vert=$ [ANS]\nCompute $\\Vert \\mathbf{v}\\Vert=$ [ANS]\nCompute $\\mathbf{u} \\cdot \\mathbf{v}=$ [ANS]", "answer_v1": [ "6.16441400296898", "7.54983443527075", "5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Distance and Dot Products: Consider the vectors {\\mathbf u}=\\langle-9, 9,-7\\rangle \\textrm{and} {\\mathbf v}=\\langle-3, 9,-4\\rangle. Compute $\\Vert \\mathbf{u}\\Vert=$ [ANS]\nCompute $\\Vert \\mathbf{v}\\Vert=$ [ANS]\nCompute $\\mathbf{u} \\cdot \\mathbf{v}=$ [ANS]", "answer_v2": [ "14.5258390463339", "10.295630140987", "136" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Distance and Dot Products: Consider the vectors {\\mathbf u}=\\langle-4, 2,-5\\rangle \\textrm{and} {\\mathbf v}=\\langle 1,-6,-3\\rangle. Compute $\\Vert \\mathbf{u}\\Vert=$ [ANS]\nCompute $\\Vert \\mathbf{v}\\Vert=$ [ANS]\nCompute $\\mathbf{u} \\cdot \\mathbf{v}=$ [ANS]", "answer_v3": [ "6.70820393249937", "6.78232998312527", "-1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0093", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "inner product", "norm" ], "problem_v1": "Find the following norms: (Type in exact answers, e.g. sqrt(2) for $\\sqrt{2}$.)\n(a) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 8 \\\\ 6 \\\\ 7 \\\\ 8\\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (b) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix}-2 \\\\-2 \\\\ 1 \\\\ 1 \\\\-1 \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (c) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 4+6i \\\\ 3+5i \\\\ 4+4i \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS].", "answer_v1": [ "14.5945", "3.31662", "10.8628" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the following norms: (Type in exact answers, e.g. sqrt(2) for $\\sqrt{2}$.)\n(a) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 1 \\\\ 10 \\\\ 2 \\\\ 4\\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (b) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 5 \\\\-2 \\\\-3 \\\\-2 \\\\ 1 \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (c) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 1+6i \\\\ 4+7i \\\\ 2+3i \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS].", "answer_v2": [ "11", "6.55744", "10.7238" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the following norms: (Type in exact answers, e.g. sqrt(2) for $\\sqrt{2}$.)\n(a) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 4 \\\\ 7 \\\\ 3 \\\\ 6\\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (b) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix}-3 \\\\-2 \\\\ 3 \\\\ 5 \\\\ 4 \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS]. (c) $\\| \\mathbf{u} \\|$ for \\mathbf{u}=\\begin{bmatrix} 2+4i \\\\ 3+2i \\\\ 5+8i \\end{bmatrix}. Answer: $\\| \\mathbf{u} \\|=$ [ANS].", "answer_v3": [ "10.4881", "7.93725", "11.0454" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0094", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "Vector", "Dot Product" ], "problem_v1": "Perform the following operations on the vectors $\\vec{u}=\\left<3,1,1\\right>$, $\\vec{v}=\\left<2,-2,-2\\right>$, and $\\vec{w}=\\left<1,1,-1\\right>$.\n$\\vec{u} \\cdot \\vec{w}=$ [ANS]\n$(\\vec{u} \\cdot \\vec{v}) \\vec{u}=$ [ANS]\n$((\\vec{w} \\cdot \\vec{w}) \\vec{u}) \\cdot \\vec{u}=$ [ANS]\n$\\vec{u} \\cdot \\vec{v}+\\vec{v} \\cdot \\vec{w}=$ [ANS]", "answer_v1": [ "3", "(6,2,2)", "33", "4" ], "answer_type_v1": [ "NV", "OL", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Perform the following operations on the vectors $\\vec{u}=\\left<-5,5,-4\\right>$, $\\vec{v}=\\left<-2,5,-2\\right>$, and $\\vec{w}=\\left<-3,-2,1\\right>$.\n$\\vec{u} \\cdot \\vec{w}=$ [ANS]\n$(\\vec{u} \\cdot \\vec{v}) \\vec{u}=$ [ANS]\n$((\\vec{w} \\cdot \\vec{w}) \\vec{u}) \\cdot \\vec{u}=$ [ANS]\n$\\vec{u} \\cdot \\vec{v}+\\vec{v} \\cdot \\vec{w}=$ [ANS]", "answer_v2": [ "1", "(-215,215,-172)", "924", "37" ], "answer_type_v2": [ "NV", "OL", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Perform the following operations on the vectors $\\vec{u}=\\left<-2,1,-2\\right>$, $\\vec{v}=\\left<1,-3,-2\\right>$, and $\\vec{w}=\\left<3,5,4\\right>$.\n$\\vec{u} \\cdot \\vec{w}=$ [ANS]\n$(\\vec{u} \\cdot \\vec{v}) \\vec{u}=$ [ANS]\n$((\\vec{w} \\cdot \\vec{w}) \\vec{u}) \\cdot \\vec{u}=$ [ANS]\n$\\vec{u} \\cdot \\vec{v}+\\vec{v} \\cdot \\vec{w}=$ [ANS]", "answer_v3": [ "-9", "(2,-1,2)", "450", "-21" ], "answer_type_v3": [ "NV", "OL", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0095", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "4", "keywords": [ "Vector", "Length" ], "problem_v1": "A boat is heading due east at 32 km/hr (relative to the water). The current is moving toward the southwest at 11 km/hr.\nGive the vector representing the actual movement of the boat. [ANS].\nHow fast is the boat moving, relative to the ground? [ANS] km/hr.\nBy what angle does the current push the boat off its due east course? Your answer should be a positive angle less than $\\pi$ radians. [ANS].", "answer_v1": [ "(24.2218,-7.77817)", "25.4401", "0.310721" ], "answer_type_v1": [ "OL", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A boat is heading due east at 21 km/hr (relative to the water). The current is moving toward the southwest at 15 km/hr.\nGive the vector representing the actual movement of the boat. [ANS].\nHow fast is the boat moving, relative to the ground? [ANS] km/hr.\nBy what angle does the current push the boat off its due east course? Your answer should be a positive angle less than $\\pi$ radians. [ANS].", "answer_v2": [ "(10.3934,-10.6066)", "14.85", "0.79555" ], "answer_type_v2": [ "OL", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A boat is heading due east at 25 km/hr (relative to the water). The current is moving toward the southwest at 11 km/hr.\nGive the vector representing the actual movement of the boat. [ANS].\nHow fast is the boat moving, relative to the ground? [ANS] km/hr.\nBy what angle does the current push the boat off its due east course? Your answer should be a positive angle less than $\\pi$ radians. [ANS].", "answer_v3": [ "(17.2218,-7.77817)", "18.8969", "0.424222" ], "answer_type_v3": [ "OL", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0096", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Consider the vector $\\vec v=\\langle 11.6,-12.4 \\rangle$. What is its magnitude? Answer: [ANS]\nWhat is its direction? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v1": [ "16.98", "0.683157", "-0.730271" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the vector $\\vec v=\\langle-18.6,3 \\rangle$. What is its magnitude? Answer: [ANS]\nWhat is its direction? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v2": [ "18.8404", "-0.987241", "0.159232" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the vector $\\vec v=\\langle-12.1,5.6 \\rangle$. What is its magnitude? Answer: [ANS]\nWhat is its direction? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v3": [ "13.333", "-0.90752", "0.420009" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0097", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "2", "keywords": [ "vectors" ], "problem_v1": "What is the unit vector for the vector $\\langle 10,3 \\rangle$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v1": [ "0.957826", "0.287348" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "What is the unit vector for the vector $\\langle-17,18 \\rangle$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v2": [ "-0.686624", "0.727013" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "What is the unit vector for the vector $\\langle-8,4 \\rangle$? Answer: $\\langle$ [ANS], [ANS] $\\rangle$", "answer_v3": [ "-0.894427", "0.447214" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0098", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Dot product, length, and unit vectors", "level": "3", "keywords": [ "vectors" ], "problem_v1": "Imagine a scene in which a birdwatcher, whose eye is located at $(3,-6,0.6)$, is watching a bird located at $(8,6,7)$. What is the vector from the birdwatcher\u2019s eye to the bird? $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Express that vector as a direction (without magnitude). $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Suppose the birdwatcher is using a telescope, which we wish to represent in a 3D scene as a cylinder extending from the birdwatcher\u2019s eye to the bird. We know the center of one end of the cylinder (the eye point), but need to compute the other. If the telescope is 0.37 units long, what is the center of the other end of the telescope? Answer: ([ANS], [ANS], [ANS])", "answer_v1": [ "5", "12", "6.4", "0.345066", "0.828158", "0.441684", "3.12767", "-5.69358", "0.763423" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Imagine a scene in which a birdwatcher, whose eye is located at $(-7,8,0.6)$, is watching a bird located at $(1,10,3)$. What is the vector from the birdwatcher\u2019s eye to the bird? $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Express that vector as a direction (without magnitude). $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Suppose the birdwatcher is using a telescope, which we wish to represent in a 3D scene as a cylinder extending from the birdwatcher\u2019s eye to the bird. We know the center of one end of the cylinder (the eye point), but need to compute the other. If the telescope is 0.25 units long, what is the center of the other end of the telescope? Answer: ([ANS], [ANS], [ANS])", "answer_v2": [ "8", "2", "2.4", "0.931493", "0.232873", "0.279448", "-6.76713", "8.05822", "0.669862" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Imagine a scene in which a birdwatcher, whose eye is located at $(-3,-7,0.7)$, is watching a bird located at $(4,7,4)$. What is the vector from the birdwatcher\u2019s eye to the bird? $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Express that vector as a direction (without magnitude). $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Suppose the birdwatcher is using a telescope, which we wish to represent in a 3D scene as a cylinder extending from the birdwatcher\u2019s eye to the bird. We know the center of one end of the cylinder (the eye point), but need to compute the other. If the telescope is 0.45 units long, what is the center of the other end of the telescope? Answer: ([ANS], [ANS], [ANS])", "answer_v3": [ "7", "14", "3.3", "0.437594", "0.875188", "0.206294", "-2.80308", "-6.60617", "0.792832" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0099", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate $\\bf{v} \\times \\bf{w}$:\n$\\mathbf{v}=\\left<6,5,5\\right>,\\quad \\mathbf{w}=\\left<6,2,2\\right>$ $\\mathbf{v}\\times \\mathbf{w}=$ [ANS]", "answer_v1": [ "0*i+18*j+-18*k" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate $\\bf{v} \\times \\bf{w}$:\n$\\mathbf{v}=\\left<0,8,1\\right>,\\quad \\mathbf{w}=\\left<3,8,2\\right>$ $\\mathbf{v}\\times \\mathbf{w}=$ [ANS]", "answer_v2": [ "8*i+3*j+-24*k" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate $\\bf{v} \\times \\bf{w}$:\n$\\mathbf{v}=\\left<2,5,2\\right>,\\quad \\mathbf{w}=\\left<4,1,3\\right>$ $\\mathbf{v}\\times \\mathbf{w}=$ [ANS]", "answer_v3": [ "13*i+2*j+-18*k" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0100", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the cross product assuming that $\\mathbf{u} \\times \\mathbf{w}=\\left< \\begin{array}{ccc} 4, & 1, & 2 \\end{array} \\right>$ $\\left(2 \\mathbf{u}-2 \\mathbf{w} \\right) \\times \\mathbf{w}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v1": [ "8", "2", "4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Calculate the cross product assuming that $\\mathbf{u} \\times \\mathbf{w}=\\left< \\begin{array}{ccc}-7, & 7, &-6 \\end{array} \\right>$ $\\left(-2 \\mathbf{u}+5 \\mathbf{w} \\right) \\times \\mathbf{w}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v2": [ "14", "-14", "12" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Calculate the cross product assuming that $\\mathbf{u} \\times \\mathbf{w}=\\left< \\begin{array}{ccc}-3, & 2, &-4 \\end{array} \\right>$ $\\left(\\mathbf{u}-3 \\mathbf{w} \\right) \\times \\mathbf{w}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v3": [ "-3", "2", "-4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0101", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Sketch the triangle with vertices $O, P=\\left(0,6,5\\right)$ and $Q=\\left(0,5,6\\right)$ and compute its area using cross products. Area=[ANS]", "answer_v1": [ "1/2*sqrt(121)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the triangle with vertices $O, P=\\left(7,0,2\\right)$ and $Q=\\left(0,2,4\\right)$ and compute its area using cross products. Area=[ANS]", "answer_v2": [ "1/2*sqrt(996)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the triangle with vertices $O, P=\\left(0,3,5\\right)$ and $Q=\\left(3,5,0\\right)$ and compute its area using cross products. Area=[ANS]", "answer_v3": [ "1/2*sqrt(931)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0102", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let $\\mathbf{v}=\\left< \\begin{array}{ccc} 2, & 1, & 1 \\end{array} \\right>$ Calculate:\n$\\mathbf{v}\\times\\bf{i}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{j}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{k}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v1": [ "0", "1", "-1", "-1", "0", "2", "1", "-2", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\mathbf{v}=\\left< \\begin{array}{ccc}-4, & 4, &-3 \\end{array} \\right>$ Calculate:\n$\\mathbf{v}\\times\\bf{i}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{j}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{k}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v2": [ "0", "-3", "-4", "3", "0", "-4", "4", "4", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\mathbf{v}=\\left< \\begin{array}{ccc}-2, & 1, &-2 \\end{array} \\right>$ Calculate:\n$\\mathbf{v}\\times\\bf{i}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{j}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$ $\\mathbf{v}\\times\\bf{k}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left.\\right>$", "answer_v3": [ "0", "-2", "-1", "2", "0", "-2", "1", "2", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0103", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Implicit", "Plane", "Perpendicular", "Normal" ], "problem_v1": "The plane that passes through the point $\\left(0,-1,-1\\right)$ and is perpendicular to both $2x-2y-2z=2$ and $2y-3z=5$ has [ANS] as its implicit equation.", "answer_v1": [ "10*x+6*y+4*z = -10" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "The plane that passes through the point $\\left(3,-3,-3\\right)$ and is perpendicular to both $5y-2x-2z=43$ and $2y-5x-z=10$ has [ANS] as its implicit equation.", "answer_v2": [ "8*y-x+21*z = -90" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "The plane that passes through the point $\\left(-5,1,5\\right)$ and is perpendicular to both $x-3y-2z=-1$ and $3x+2y+3z=31$ has [ANS] as its implicit equation.", "answer_v3": [ "5*x+9*y-11*z = -71" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Geometry_0104", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Vector", "Angle", "Dot Product", "Cross Product" ], "problem_v1": "Suppose $\\overline{u}=\\left<4,1,1\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<3,-4,-3\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<1,3,2\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<2,2,-2\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<-2,0,8\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline\\end{array}$", "answer_v1": [ "an acute angle", "an acute angle", "an acute angle", "a right angle" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ], "problem_v2": "Suppose $\\overline{u}=\\left<1,5,-4\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<8,0,2\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<-2,5,-2\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<-3,2,3\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<1,-4,-1\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline\\end{array}$", "answer_v2": [ "a right angle", "an acute angle", "an obtuse angle", "an obtuse angle" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ], "problem_v3": "Suppose $\\overline{u}=\\left<2,1,-2\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<1,2,-2\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<2,-2,-3\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<4,0,4\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline \\\\\\hline \\left<3,5,5\\right>makes & & [ANS]an acute anglewith \\overline{u} \\\\\\hline\\end{array}$", "answer_v3": [ "an acute angle", "an acute angle", "a right angle", "an acute angle" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ] }, { "id": "Geometry_0105", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Vector", "Perpendicular", "Cross Product" ], "problem_v1": "Let $A=\\left(3,1,1\\right)$, $B=\\left(2,-2,-2\\right)$, and $P=(k,k,k)$. The vector from $A$ to $B$ is perpendicular to the vector from $A$ to $P$ when $k$=[ANS].", "answer_v1": [ "1.28571" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $A=\\left(-5,5,-4\\right)$, $B=\\left(-2,5,-2\\right)$, and $P=(k,k,k)$. The vector from $A$ to $B$ is perpendicular to the vector from $A$ to $P$ when $k$=[ANS].", "answer_v2": [ "-4.6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $A=\\left(-2,1,-2\\right)$, $B=\\left(1,-3,-2\\right)$, and $P=(k,k,k)$. The vector from $A$ to $B$ is perpendicular to the vector from $A$ to $P$ when $k$=[ANS].", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0106", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Vector", "Parallel", "Perpendicular", "Cross Product", "Dot Product" ], "problem_v1": "Suppose $\\overline{v}=\\left<3,1,1\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<4.5,1.5,1.5\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<-6,-2,-2\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<0,8,-8\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<-2,4,2\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline\\end{array}$", "answer_v1": [ "parallel", "parallel", "perpendicular", "perpendicular" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v2": "Suppose $\\overline{v}=\\left<-5,5,-4\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<-3,17,25\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<10,-10,8\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<10,-2,-15\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<-7.5,7.5,-6\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline\\end{array}$", "answer_v2": [ "perpendicular", "parallel", "perpendicular", "parallel" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v3": "Suppose $\\overline{v}=\\left<-2,1,-2\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<-3,-2,-3\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<4,-2,4\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<-8,-6,5\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline \\\\\\hline \\left<14,2,-13\\right>is & & [ANS]neitherto \\overline{v} \\\\\\hline\\end{array}$", "answer_v3": [ "neither", "parallel", "perpendicular", "perpendicular" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ] }, { "id": "Geometry_0107", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "", "keywords": [], "problem_v1": "Find the volume of the parallelepiped defined by the vectors \\left[\\begin{array}{c} 2\\cr 1\\cr 2\\cr 3 \\end{array}\\right], \\left[\\begin{array}{c}-2\\cr-2\\cr 2\\cr 2 \\end{array}\\right], \\left[\\begin{array}{c}-2\\cr 2\\cr 1\\cr-2 \\end{array}\\right]. Volume=[ANS].", "answer_v1": [ "sqrt(2984)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the parallelepiped defined by the vectors \\left[\\begin{array}{c} 0\\cr 2\\cr 1\\cr 2 \\end{array}\\right], \\left[\\begin{array}{c}-1\\cr-2\\cr 1\\cr 1 \\end{array}\\right], \\left[\\begin{array}{c}-2\\cr 1\\cr 1\\cr-1 \\end{array}\\right]. Volume=[ANS].", "answer_v2": [ "sqrt(427)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the parallelepiped defined by the vectors \\left[\\begin{array}{c} 0\\cr 1\\cr 1\\cr 2 \\end{array}\\right], \\left[\\begin{array}{c}-2\\cr-2\\cr 2\\cr 2 \\end{array}\\right], \\left[\\begin{array}{c}-1\\cr 1\\cr 0\\cr-2 \\end{array}\\right]. Volume=[ANS].", "answer_v3": [ "sqrt(336)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0108", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "", "keywords": [], "problem_v1": "Find the area of the parallelogram defined by the vectors \\left[\\begin{array}{c} 7\\cr 6 \\end{array}\\right], \\left[\\begin{array}{c}-4\\cr 7 \\end{array}\\right]. Area=[ANS].", "answer_v1": [ "7*7-6*-4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the parallelogram defined by the vectors \\left[\\begin{array}{c} 1\\cr 9 \\end{array}\\right], \\left[\\begin{array}{c}-8\\cr 4 \\end{array}\\right]. Area=[ANS].", "answer_v2": [ "1*4-9*-8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the parallelogram defined by the vectors \\left[\\begin{array}{c} 3\\cr 6 \\end{array}\\right], \\left[\\begin{array}{c}-7\\cr 5 \\end{array}\\right]. Area=[ANS].", "answer_v3": [ "3*5-6*-7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0109", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "", "keywords": [], "problem_v1": "Find the volume of the tetrahedron with vertices $(0, 2,-3)$, $(3, 4,-2)$, $(1, 0,-2)$, and $(1, 0,-4)$.\nVolume=[ANS].", "answer_v1": [ "16/6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the tetrahedron with vertices $(-5, 2,-1)$, $(-10, 0,-4)$, $(0, 7,-3)$, and $(-9, 0, 0)$.\nVolume=[ANS].", "answer_v2": [ "41/6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the tetrahedron with vertices $(-3,-2,-3)$, $(-5,-1, 0)$, $(-2,-5, 2)$, and $(-5,-4, 1)$.\nVolume=[ANS].", "answer_v3": [ "34/6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0110", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "Cross Product", "Vector" ], "problem_v1": "If $\\mathbf{a}=\\mathbf{i}+\\mathbf{j}+5 \\mathbf{k}$ and $\\mathbf{b}=\\mathbf{i}+\\mathbf{j}+4 \\mathbf{k}$ Compute the cross product $\\bf{a} \\times \\bf{b}$.\n$\\bf{a} \\times \\bf{b}=$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v1": [ "-1", "1", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $\\mathbf{a}=\\mathbf{i}+\\mathbf{j}+2 \\mathbf{k}$ and $\\mathbf{b}=\\mathbf{i}+\\mathbf{j}+5 \\mathbf{k}$ Compute the cross product $\\bf{a} \\times \\bf{b}$.\n$\\bf{a} \\times \\bf{b}=$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v2": [ "3", "-3", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $\\mathbf{a}=\\mathbf{i}+\\mathbf{j}+3 \\mathbf{k}$ and $\\mathbf{b}=\\mathbf{i}+\\mathbf{j}+4 \\mathbf{k}$ Compute the cross product $\\bf{a} \\times \\bf{b}$.\n$\\bf{a} \\times \\bf{b}=$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v3": [ "1", "-1", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0111", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "Cross Product", "Vector" ], "problem_v1": "Let $\\mathbf{a}=(8, 6, 7)$ and $\\mathbf{b}=(8, 4, 4)$ be vectors. Compute the cross product $\\mathbf{a} \\times \\mathbf{b}$.\n$\\mathbf{a} \\times \\mathbf{b}=$ ([ANS], [ANS], [ANS])", "answer_v1": [ "-4", "24", "-16" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\mathbf{a}=(1, 10, 2)$ and $\\mathbf{b}=(4, 10, 4)$ be vectors. Compute the cross product $\\mathbf{a} \\times \\mathbf{b}$.\n$\\mathbf{a} \\times \\mathbf{b}=$ ([ANS], [ANS], [ANS])", "answer_v2": [ "20", "4", "-30" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\mathbf{a}=(4, 7, 3)$ and $\\mathbf{b}=(6, 3, 4)$ be vectors. Compute the cross product $\\mathbf{a} \\times \\mathbf{b}$.\n$\\mathbf{a} \\times \\mathbf{b}=$ ([ANS], [ANS], [ANS])", "answer_v3": [ "19", "2", "-30" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0112", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Cross Product", "Vector" ], "problem_v1": "You are looking down at a map. A vector $\\bf{u}$ with $\\left| \\mathbf{u} \\right|$=8 points north and a vector $\\mathbf{v}$ with $\\left| \\mathbf{v} \\right|$=6 points northeast. The crossproduct $\\mathbf{u} \\times \\mathbf{v}$ points: A) south B) northwest C) up D) down\nPlease enter the letter of the correct answer: [ANS]\nThe magnitude $\\left| \\mathbf{u} \\times \\mathbf{v} \\right|$=[ANS]", "answer_v1": [ "D", "(8*6*sqrt(2) )/2" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You are looking down at a map. A vector $\\bf{u}$ with $\\left| \\mathbf{u} \\right|$=1 points north and a vector $\\mathbf{v}$ with $\\left| \\mathbf{v} \\right|$=10 points northeast. The crossproduct $\\mathbf{u} \\times \\mathbf{v}$ points: A) south B) northwest C) up D) down\nPlease enter the letter of the correct answer: [ANS]\nThe magnitude $\\left| \\mathbf{u} \\times \\mathbf{v} \\right|$=[ANS]", "answer_v2": [ "D", "(1*10*sqrt(2) )/2" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You are looking down at a map. A vector $\\bf{u}$ with $\\left| \\mathbf{u} \\right|$=4 points north and a vector $\\mathbf{v}$ with $\\left| \\mathbf{v} \\right|$=7 points northeast. The crossproduct $\\mathbf{u} \\times \\mathbf{v}$ points: A) south B) northwest C) up D) down\nPlease enter the letter of the correct answer: [ANS]\nThe magnitude $\\left| \\mathbf{u} \\times \\mathbf{v} \\right|$=[ANS]", "answer_v3": [ "D", "(4*7*sqrt(2) )/2" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0113", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Angle", "Plane" ], "problem_v1": "A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (3, 2, 0), and (0, 2, 3). By what angle does the tower now deviate from the vertical? [ANS] radians.", "answer_v1": [ "1.0643516833814" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (1, 3, 0), and (0, 1, 2). By what angle does the tower now deviate from the vertical? [ANS] radians.", "answer_v2": [ "1.4139806414505" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (1, 2, 0), and (0, 1, 2). By what angle does the tower now deviate from the vertical? [ANS] radians.", "answer_v3": [ "1.35080834939944" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0114", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "4", "keywords": [ "Torque", "Newton", "Force", "Force", "Torque", "Cross Product" ], "problem_v1": "A bicycle pedal is pushed straight downwards by a foot with a 40 Newton force. The shaft of the pedal is 20 cm long. If the shaft is $\\pi/5$ radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? [ANS] Nm", "answer_v1": [ "(2/10) * 40 * sin(pi/2 - pi/5 )" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A bicycle pedal is pushed straight downwards by a foot with a 13 Newton force. The shaft of the pedal is 20 cm long. If the shaft is $\\pi/6$ radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? [ANS] Nm", "answer_v2": [ "(2/10) * 13 * sin(pi/2 - pi/6 )" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A bicycle pedal is pushed straight downwards by a foot with a 22 Newton force. The shaft of the pedal is 20 cm long. If the shaft is $\\pi/5$ radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? [ANS] Nm", "answer_v3": [ "(2/10) * 22 * sin(pi/2 - pi/5 )" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0115", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "cross product", "vectors" ], "problem_v1": "If $\\vec v \\times \\vec w=4\\mathit{\\vec i}+\\mathit{\\vec j}+\\mathit{\\vec k}$, and $\\vec v \\cdot \\vec w=4$, and $\\theta$ is the angle between $\\vec v$ and $\\vec w$, then\n(a) $\\tan \\theta=$ [ANS]\n(b) $\\theta=$ [ANS]", "answer_v1": [ "4.24264/4", "atan(4.24264/4)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $\\vec v \\times \\vec w=\\mathit{\\vec i}+3\\mathit{\\vec j}-4\\mathit{\\vec k}$, and $\\vec v \\cdot \\vec w=3$, and $\\theta$ is the angle between $\\vec v$ and $\\vec w$, then\n(a) $\\tan \\theta=$ [ANS]\n(b) $\\theta=$ [ANS]", "answer_v2": [ "5.09902/3", "atan(5.09902/3)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $\\vec v \\times \\vec w=2\\mathit{\\vec i}+\\mathit{\\vec j}-2\\mathit{\\vec k}$, and $\\vec v \\cdot \\vec w=4$, and $\\theta$ is the angle between $\\vec v$ and $\\vec w$, then\n(a) $\\tan \\theta=$ [ANS]\n(b) $\\theta=$ [ANS]", "answer_v3": [ "3/4", "atan(3/4)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0116", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "4", "keywords": [ "cross product", "vectors" ], "problem_v1": "In the following, find the vector representing the area of a surface. The magnitude of the vector equals the magnitude of the area; the direction is perpendicular to the surface. Since there are two perpendicular directions, we pick one by giving an orientation for the surface.\n(a) The area vector $\\vec{A_1}$ for the rectangle with vertices $(0,0,0)$, $(2,0,0)$, $(2,1,0)$, and $(0,1,0)$, oriented so that it faces downward. $\\vec{A_1}=$ [ANS]\n(b) The area vector $\\vec{A_2}$ for a circle of radius 3 in the $yz$-plane, facing in the direction of the positive $x$-axis. $\\vec{A_2}=$ [ANS]", "answer_v1": [ "-2k", "28.2743i" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "In the following, find the vector representing the area of a surface. The magnitude of the vector equals the magnitude of the area; the direction is perpendicular to the surface. Since there are two perpendicular directions, we pick one by giving an orientation for the surface.\n(a) The area vector $\\vec{A_1}$ for the rectangle with vertices $(0,0,0)$, $(-3,0,0)$, $(-3,3,0)$, and $(0,3,0)$, oriented so that it faces downward. $\\vec{A_1}=$ [ANS]\n(b) The area vector $\\vec{A_2}$ for a circle of radius 2 in the $xz$-plane, facing in the direction of the positive $y$-axis. $\\vec{A_2}=$ [ANS]", "answer_v2": [ "-9k", "12.5664j" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "In the following, find the vector representing the area of a surface. The magnitude of the vector equals the magnitude of the area; the direction is perpendicular to the surface. Since there are two perpendicular directions, we pick one by giving an orientation for the surface.\n(a) The area vector $\\vec{A_1}$ for the rectangle with vertices $(0,0,0)$, $(-1,0,0)$, $(-1,1,0)$, and $(0,1,0)$, oriented so that it faces downward. $\\vec{A_1}=$ [ANS]\n(b) The area vector $\\vec{A_2}$ for a circle of radius 2 in the $yz$-plane, facing in the direction of the positive $x$-axis. $\\vec{A_2}=$ [ANS]", "answer_v3": [ "-k", "12.5664i" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0117", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "cross product", "vectors" ], "problem_v1": "Use the algebraic definition to find $\\vec v \\times \\vec w$ if $\\vec{v}=2\\mathit{\\vec i}+\\mathit{\\vec j}+\\mathit{\\vec k}$ and $\\vec{w}=2\\mathit{\\vec i}-\\mathit{\\vec j}-\\mathit{\\vec k}$. $\\vec v \\times \\vec w=$ [ANS]", "answer_v1": [ "4j-4k" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the algebraic definition to find $\\vec v \\times \\vec w$ if $\\vec{v}=-3\\mathit{\\vec i}+3\\mathit{\\vec j}-2\\mathit{\\vec k}$ and $\\vec{w}=-\\mathit{\\vec i}+3\\mathit{\\vec j}-\\mathit{\\vec k}$. $\\vec v \\times \\vec w=$ [ANS]", "answer_v2": [ "3i-j-6k" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the algebraic definition to find $\\vec v \\times \\vec w$ if $\\vec{v}=-\\mathit{\\vec i}+\\mathit{\\vec j}-2\\mathit{\\vec k}$ and $\\vec{w}=-2\\mathit{\\vec i}-\\mathit{\\vec j}+2\\mathit{\\vec k}$. $\\vec v \\times \\vec w=$ [ANS]", "answer_v3": [ "6j+3k" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0118", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "2", "keywords": [ "Vector", "The Cross Product" ], "problem_v1": "Use the geometric definition of the cross product and the properties of the cross product to make the following calculations.\n(a) $((\\vec{j}+\\vec{k}) \\times \\vec{j}) \\times \\vec{k}$=[ANS]\n(b) $(\\vec{i}+\\vec{j}) \\times (\\vec{i} \\times \\vec{j})$=[ANS]\n(c) $4 \\vec{k} \\times (\\vec{k}+\\vec{j})$=[ANS]\n(d) $(\\vec{i}+\\vec{j}) \\times (\\vec{i}-\\vec{j})$=[ANS]", "answer_v1": [ "(0,1,0)", "(1,-1,0)", "(-4,0,0)", "(0,0,-2)" ], "answer_type_v1": [ "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use the geometric definition of the cross product and the properties of the cross product to make the following calculations.\n(a) $((\\vec{i}+\\vec{j}) \\times \\vec{i}) \\times \\vec{j}$=[ANS]\n(b) $(\\vec{j}+\\vec{k}) \\times (\\vec{j} \\times \\vec{k})$=[ANS]\n(c) $5 \\vec{i} \\times (\\vec{i}+\\vec{j})$=[ANS]\n(d) $(\\vec{k}+\\vec{j}) \\times (\\vec{k}-\\vec{j})$=[ANS]", "answer_v2": [ "(1,0,0)", "(0,1,-1)", "(0,0,5)", "(2,0,0)" ], "answer_type_v2": [ "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use the geometric definition of the cross product and the properties of the cross product to make the following calculations.\n(a) $((\\vec{i}+\\vec{j}) \\times \\vec{i}) \\times \\vec{j}$=[ANS]\n(b) $(\\vec{j}+\\vec{k}) \\times (\\vec{j} \\times \\vec{k})$=[ANS]\n(c) $4 \\vec{i} \\times (\\vec{i}+\\vec{j})$=[ANS]\n(d) $(\\vec{k}+\\vec{j}) \\times (\\vec{k}-\\vec{j})$=[ANS]", "answer_v3": [ "(1,0,0)", "(0,1,-1)", "(0,0,4)", "(2,0,0)" ], "answer_type_v3": [ "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0119", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Vector", "The Cross Product" ], "problem_v1": "Let $A=(1, 1,-1)$, $B=(4, 2, 0)$, $C=(6, 0,-2)$, and $D=(3,-1,-3)$. Find the area of the parallelogram determined by these four points, the area of the triangle $ABC$, and the area of the triangle $ABD$.\nArea of parallelogram $ABCD$=[ANS]\nArea of triangle $ABC$=[ANS]\nArea of triangle $ABD$=[ANS]", "answer_v1": [ "11.3137", "5.65685", "5.65685" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $A=(-3,-2, 1)$, $B=(-8, 3,-3)$, $C=(-10, 8,-5)$, and $D=(-5, 3,-1)$. Find the area of the parallelogram determined by these four points, the area of the triangle $ABC$, and the area of the triangle $ABD$.\nArea of parallelogram $ABCD$=[ANS]\nArea of triangle $ABC$=[ANS]\nArea of triangle $ABD$=[ANS]", "answer_v2": [ "18.1384", "9.06918", "9.06918" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $A=(3, 5, 4)$, $B=(1, 6, 2)$, $C=(2, 3, 0)$, and $D=(4, 2, 2)$. Find the area of the parallelogram determined by these four points, the area of the triangle $ABC$, and the area of the triangle $ABD$.\nArea of parallelogram $ABCD$=[ANS]\nArea of triangle $ABC$=[ANS]\nArea of triangle $ABD$=[ANS]", "answer_v3": [ "11.1803", "5.59017", "5.59017" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0120", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "Vector", "Length" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. For any scalar $c$ and any vector $\\vec{v}$, we have $||c\\vec{v}||=c ||\\vec{v}||$. [ANS] 2. $(\\vec{i} \\times \\vec{j}) \\cdot \\vec{k}=\\vec{i} \\cdot (\\vec{j} \\times \\vec{k})$. [ANS] 3. The value of $\\vec{v} \\cdot (\\vec{v} \\times \\vec{w})$ is always zero. [ANS] 4. If $\\vec{v}$ and $\\vec{w}$ are any two vectors, then $||\\vec{v}+\\vec{w}||=||\\vec{v}||+||\\vec{w}||$.", "answer_v1": [ "FALSE", "TRUE", "TRUE", "False" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. $(\\vec{i} \\times \\vec{j}) \\cdot \\vec{k}=\\vec{i} \\cdot (\\vec{j} \\times \\vec{k})$. [ANS] 2. The value of $\\vec{v} \\cdot (\\vec{v} \\times \\vec{w})$ is always zero. [ANS] 3. For any scalar $c$ and any vector $\\vec{v}$, we have $||c\\vec{v}||=c ||\\vec{v}||$. [ANS] 4. If $\\vec{v}$ and $\\vec{w}$ are any two vectors, then $||\\vec{v}+\\vec{w}||=||\\vec{v}||+||\\vec{w}||$.", "answer_v2": [ "TRUE", "TRUE", "FALSE", "False" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. $(\\vec{i} \\times \\vec{j}) \\cdot \\vec{k}=\\vec{i} \\cdot (\\vec{j} \\times \\vec{k})$. [ANS] 2. If $\\vec{v}$ and $\\vec{w}$ are any two vectors, then $||\\vec{v}+\\vec{w}||=||\\vec{v}||+||\\vec{w}||$. [ANS] 3. For any scalar $c$ and any vector $\\vec{v}$, we have $||c\\vec{v}||=c ||\\vec{v}||$. [ANS] 4. The value of $\\vec{v} \\cdot (\\vec{v} \\times \\vec{w})$ is always zero.", "answer_v3": [ "TRUE", "FALSE", "FALSE", "True" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Geometry_0121", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "cross product' 'area" ], "problem_v1": "Find the area of the triangle with vertices $(0, 0, 0), (3, 1, 1),$ and $(3, 2, 0).$\n$A=$ [ANS]", "answer_v1": [ "2.34520787991171" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the triangle with vertices $(0, 0, 0), (-5, 5,-4),$ and $(-5, 4,-2).$\n$A=$ [ANS]", "answer_v2": [ "6.34428877022476" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the triangle with vertices $(0, 0, 0), (-2, 1,-2),$ and $(-2, 0,-3).$\n$A=$ [ANS]", "answer_v3": [ "2.06155281280883" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0122", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "3", "keywords": [ "cross product", "vectors" ], "problem_v1": "Let $P=(0,0,0), Q=(1,1,1), R=(-1,-1,-1)$. Find the area of the triangle $PQR$. area=[ANS]\nLet $T=(-4,1,2), U=(-3,3,-5), V=(-2,-2,-1)$. Find the area of the triangle $TUV$. area=[ANS]", "answer_v1": [ "0", "14.9916643505649" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $P=(0,0,0), Q=(1,-1,2), R=(-2,1,1)$. Find the area of the triangle $PQR$. area=[ANS]\nLet $T=(9,-4,-7), U=(-4,1,-9), V=(3,-1,7)$. Find the area of the triangle $TUV$. area=[ANS]", "answer_v2": [ "2.95803989154981", "104.274877127715" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $P=(0,0,0), Q=(1,-1,1), R=(-1,1,-1)$. Find the area of the triangle $PQR$. area=[ANS]\nLet $T=(-6,-3,6), U=(9,8,-6), V=(-4,-5,-10)$. Find the area of the triangle $TUV$. area=[ANS]", "answer_v3": [ "0", "149.465715132267" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0123", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Cross product", "level": "4", "keywords": [ "cross product", "vectors" ], "problem_v1": "A triangle is defined by the points $A(5,2,3)$, $B(5,-4,-4)$, and $C(1,k,-3)$. The area of the triangle is $\\sqrt{664}$. Determine the value of k.\nk=[ANS] or k=[ANS]", "answer_v1": [ "-8.28571428571429", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A triangle is defined by the points $A(-9,9,-7)$, $B(-3,9,-4)$, and $C(-7,k,1)$. The area of the triangle is $\\sqrt{2342.25}$. Determine the value of k.\nk=[ANS] or k=[ANS]", "answer_v2": [ "-4", "22" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A triangle is defined by the points $A(-4,2,-5)$, $B(1,-6,-3)$, and $C(6,k,8)$. The area of the triangle is $\\sqrt{7293.5}$. Determine the value of k.\nk=[ANS] or k=[ANS]", "answer_v3": [ "-46.9310344827586", "9" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0124", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find a parametrization of the line through the origin whose projection on the $xy$-plane is a line of slope 8 and on the $yz$-plane is a line of slope 6 $($ i.e., $\\Delta z/\\Delta y=6)$. Scale your answer so that the smallest coefficient is 1. $\\mathbf r (t)=\\langle$ [ANS] $t,$ [ANS] $t,$ [ANS] $t \\rangle$", "answer_v1": [ "1", "8", "48" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a parametrization of the line through the origin whose projection on the $xy$-plane is a line of slope 2 and on the $yz$-plane is a line of slope 9 $($ i.e., $\\Delta z/\\Delta y=9)$. Scale your answer so that the smallest coefficient is 1. $\\mathbf r (t)=\\langle$ [ANS] $t,$ [ANS] $t,$ [ANS] $t \\rangle$", "answer_v2": [ "1", "2", "18" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a parametrization of the line through the origin whose projection on the $xy$-plane is a line of slope 4 and on the $yz$-plane is a line of slope 6 $($ i.e., $\\Delta z/\\Delta y=6)$. Scale your answer so that the smallest coefficient is 1. $\\mathbf r (t)=\\langle$ [ANS] $t,$ [ANS] $t,$ [ANS] $t \\rangle$", "answer_v3": [ "1", "4", "24" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0125", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "1", "keywords": [ "calculus" ], "problem_v1": "Find a vector parametrization of the line through $P=\\left(5,2,2 \\right)$ in the direction ${\\bf v}=\\left< 4,-4,-3 \\right>$ $\\bf r\\it (t)=$ ([ANS] $+$ [ANS] $t$) ${\\bf i}+$ ([ANS] $+$ [ANS] $t$) ${\\bf j}+$ ([ANS] $+$ [ANS] $t$) ${\\bf k}$", "answer_v1": [ "5", "4", "2", "-4", "2", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find a vector parametrization of the line through $P=\\left(-8,8,-7 \\right)$ in the direction ${\\bf v}=\\left<-3,8,-3 \\right>$ $\\bf r\\it (t)=$ ([ANS] $+$ [ANS] $t$) ${\\bf i}+$ ([ANS] $+$ [ANS] $t$) ${\\bf j}+$ ([ANS] $+$ [ANS] $t$) ${\\bf k}$", "answer_v2": [ "-8", "-3", "8", "8", "-7", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find a vector parametrization of the line through $P=\\left(-4,2,-4 \\right)$ in the direction ${\\bf v}=\\left< 1,-6,-3 \\right>$ $\\bf r\\it (t)=$ ([ANS] $+$ [ANS] $t$) ${\\bf i}+$ ([ANS] $+$ [ANS] $t$) ${\\bf j}+$ ([ANS] $+$ [ANS] $t$) ${\\bf k}$", "answer_v3": [ "-4", "1", "2", "-6", "-4", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Geometry_0128", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "2", "keywords": [ "Line", "Parametric", "Parallel", "Perpendicular" ], "problem_v1": "Consider the line $L(t)=\\left<1+3t,2+t\\right>$. Then:\n$\\begin{array}{cccc}\\hline & Lis & & [ANS]neitherto the line \\left<2-6t,-3-2t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<-1,-1-t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<1+3t,1-9t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<-2-2t,6t-2\\right> \\\\\\hline\\end{array}$", "answer_v1": [ "parallel", "neither", "perpendicular", "perpendicular" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v2": "Consider the line $L(t)=\\left<-4-5t,5t-2\\right>$. Then:\n$\\begin{array}{cccc}\\hline & Lis & & [ANS]neitherto the line \\left<5-10t,-2-10t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<15t-3,15t-2\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<1-7.5t,7.5t-5\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<2+10t,-1-10t\\right> \\\\\\hline\\end{array}$", "answer_v2": [ "perpendicular", "perpendicular", "parallel", "parallel" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v3": "Consider the line $L(t)=\\left<-2-2t,1+t\\right>$. Then:\n$\\begin{array}{cccc}\\hline & Lis & & [ANS]neitherto the line \\left<5-5t,3+t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<4-3t,1.5t-3\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left<3+3t,5+6t\\right> \\\\\\hline \\\\\\hline & Lis & & [ANS]neitherto the line \\left \\\\\\hline\\end{array}$", "answer_v3": [ "neither", "parallel", "perpendicular", "neither" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ] }, { "id": "Geometry_0129", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "Parametric", "Line", "Vector" ], "problem_v1": "Suppose a line is given by the system of equations $x=3+2t$, $y=1-2t$, $z=1-2t$. Then the vector and point that were used to define this line were $\\overline{v}$=[ANS], and $p$=[ANS].", "answer_v1": [ "(2,-2,-2)", "(3,1,1)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Suppose a line is given by the system of equations $x=-5-2t$, $y=5+5t$, $z=-4-2t$. Then the vector and point that were used to define this line were $\\overline{v}$=[ANS], and $p$=[ANS].", "answer_v2": [ "(-2,5,-2)", "(-5,5,-4)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Suppose a line is given by the system of equations $x=t-2$, $y=1-3t$, $z=-2-2t$. Then the vector and point that were used to define this line were $\\overline{v}$=[ANS], and $p$=[ANS].", "answer_v3": [ "(1,-3,-2)", "(-2,1,-2)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0130", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "Line", "Symmetric Equation", "Vector", "Symmetric", "Multivariable", "Geometry" ], "problem_v1": "Given a the vector equation $\\mathbf{r} (t)=(3+2\\!t) \\mathbf{i}+(1-2\\!t) \\mathbf{j}+(1-2\\!t) \\mathbf{k}$, rewrite this in terms of the symmetric equations for the line. (quotient involving x) [ANS]\n(quotient involving y)=[ANS]\n(quotient involving z)=[ANS]", "answer_v1": [ "(x - 3)/2", "(y - 1)/-2", "(z - 1)/-2" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Given a the vector equation $\\mathbf{r} (t)=(-5-2\\!t) \\mathbf{i}+(5+5\\!t) \\mathbf{j}+(-4-2\\!t) \\mathbf{k}$, rewrite this in terms of the symmetric equations for the line. (quotient involving x) [ANS]\n(quotient involving y)=[ANS]\n(quotient involving z)=[ANS]", "answer_v2": [ "(x - -5)/-2", "(y - 5)/5", "(z - -4)/-2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Given a the vector equation $\\mathbf{r} (t)=(-2+1\\!t) \\mathbf{i}+(1-3\\!t) \\mathbf{j}+(-2-2\\!t) \\mathbf{k}$, rewrite this in terms of the symmetric equations for the line. (quotient involving x) [ANS]\n(quotient involving y)=[ANS]\n(quotient involving z)=[ANS]", "answer_v3": [ "(x - -2)/1", "(y - 1)/-3", "(z - -2)/-2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0131", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "4", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Is the point $(-2,-3, 2)$ visible from the point $(5, 6, 0)$ if there is an opaque ball of radius $1$ centered at the origin? [ANS] Suppose that you stand at the point $(5, 6, 0)$ and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere. If $(-2,-3, 2)$ is not visible from $(5, 6, 0)$, find the point on the sphere at which you are looking if you look in the direction of $(-2,-3, 2)$. Otherwise, find the point on the sphere at which you look if you are looking in the direction of $(-2,-3, 1)$. Point $(x, y, z)=$ [ANS]", "answer_v1": [ "yes", "(0.661616,0.422078,0.619769)" ], "answer_type_v1": [ "MCS", "OL" ], "options_v1": [ [ "yes", "no" ], [] ], "problem_v2": "Is the point $(-1,-2, 2)$ visible from the point $(1, 2, 0)$ if there is an opaque ball of radius $1$ centered at the origin? [ANS] Suppose that you stand at the point $(1, 2, 0)$ and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere. If $(-1,-2, 2)$ is not visible from $(1, 2, 0)$, find the point on the sphere at which you are looking if you look in the direction of $(-1,-2, 2)$. Otherwise, find the point on the sphere at which you look if you are looking in the direction of $(-1,-2, 1)$. Point $(x, y, z)=$ [ANS]", "answer_v2": [ "no", "(0.333333,0.666667,0.666667)" ], "answer_type_v2": [ "MCS", "OL" ], "options_v2": [ [ "yes", "no" ], [] ], "problem_v3": "Is the point $(-2,-3, 3)$ visible from the point $(2, 3, 0)$ if there is an opaque ball of radius $1$ centered at the origin? [ANS] Suppose that you stand at the point $(2, 3, 0)$ and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere. If $(-2,-3, 3)$ is not visible from $(2, 3, 0)$, find the point on the sphere at which you are looking if you look in the direction of $(-2,-3, 3)$. Otherwise, find the point on the sphere at which you look if you are looking in the direction of $(-2,-3, 2)$. Point $(x, y, z)=$ [ANS]", "answer_v3": [ "yes", "(0.285714,0.428571,0.857143)" ], "answer_type_v3": [ "MCS", "OL" ], "options_v3": [ [ "yes", "no" ], [] ] }, { "id": "Geometry_0132", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "Parametric", "Line", "Vector" ], "problem_v1": "Find the distance between the skew lines $P(t)=\\left(3,1,1\\right)+t \\left<1,1,-1\\right>$ and $Q(t)=\\left(2,-2,-2\\right)+t \\left<0,2,-3\\right>$. Hint: Take the cross product of the slope vectors of $P$ and $Q$ to find a vector normal to both of these lines.\ndistance=[ANS].", "answer_v1": [ "3.74166" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance between the skew lines $P(t)=\\left(-5,5,-4\\right)+t \\left<-3,-2,1\\right>$ and $Q(t)=\\left(-2,5,-2\\right)+t \\left<-5,2,-1\\right>$. Hint: Take the cross product of the slope vectors of $P$ and $Q$ to find a vector normal to both of these lines.\ndistance=[ANS].", "answer_v2": [ "1.78885" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance between the skew lines $P(t)=\\left(-2,1,-2\\right)+t \\left<3,5,4\\right>$ and $Q(t)=\\left(1,-3,-2\\right)+t \\left<-3,-2,-3\\right>$. Hint: Take the cross product of the slope vectors of $P$ and $Q$ to find a vector normal to both of these lines.\ndistance=[ANS].", "answer_v3": [ "0.76337" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0133", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "vectors' 'line" ], "problem_v1": "Vectors: Find the vector from the point $(3, 2, 7)$ to the point $(8,-2,-4)$. Enter the coordinates: ([ANS], [ANS], [ANS])\nEquations of lines: Consider the vector equation of the line through the two points listed above. For each equation listed below, answer T if the equation represents the line, and F if it does not.\nHere is the list of questions: [ANS] 1. $(x,y,z)=(3, 2, 7)+t\\, (8,-2,-4)$ [ANS] 2. $(x,y,z)=(8,-2,-4)+t\\, (3, 2, 7)$ [ANS] 3. $(x,y,z)=(8,-2,-4)+t\\, (5,-4,-11)$ [ANS] 4. $(x,y,z)=(3, 2, 7)+t\\, (5,-4,-11)$ [ANS] 5. $(x,y,z)=(3, 2, 7)+t\\, (-5,+4,+11)$", "answer_v1": [ "5", "-4", "-11", "F", "F", "T", "T", "T" ], "answer_type_v1": [ "NV", "NV", "NV", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Vectors: Find the vector from the point $(-5, 9, 2)$ to the point $(4, 5,-4)$. Enter the coordinates: ([ANS], [ANS], [ANS])\nEquations of lines: Consider the vector equation of the line through the two points listed above. For each equation listed below, answer T if the equation represents the line, and F if it does not.\nHere is the list of questions: [ANS] 1. $(x,y,z)=(-5, 9, 2)+t\\, (9,-4,-6)$ [ANS] 2. $(x,y,z)=(-5, 9, 2)+t\\, (4, 5,-4)$ [ANS] 3. $(x,y,z)=(-5, 9, 2)+t\\, (-9,+4,+6)$ [ANS] 4. $(x,y,z)=(4, 5,-4)+t\\, (-5, 9, 2)$ [ANS] 5. $(x,y,z)=(4, 5,-4)+t\\, (9,-4,-6)$", "answer_v2": [ "9", "-4", "-6", "T", "F", "T", "F", "T" ], "answer_type_v2": [ "NV", "NV", "NV", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Vectors: Find the vector from the point $(-2, 2, 3)$ to the point $(6,-3,-3)$. Enter the coordinates: ([ANS], [ANS], [ANS])\nEquations of lines: Consider the vector equation of the line through the two points listed above. For each equation listed below, answer T if the equation represents the line, and F if it does not.\nHere is the list of questions: [ANS] 1. $(x,y,z)=(-2, 2, 3)+t\\, (6,-3,-3)$ [ANS] 2. $(x,y,z)=(6,-3,-3)+t\\, (8,-5,-6)$ [ANS] 3. $(x,y,z)=(-2, 2, 3)+t\\, (8,-5,-6)$ [ANS] 4. $(x,y,z)=(-2, 2, 3)+t\\, (-8,+5,+6)$ [ANS] 5. $(x,y,z)=(6,-3,-3)+t\\, (-2, 2, 3)$", "answer_v3": [ "8", "-5", "-6", "F", "T", "T", "T", "F" ], "answer_type_v3": [ "NV", "NV", "NV", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0134", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "calculus", "line", "plane" ], "problem_v1": "Determine whether the lines L_1: x=21+6 t, \\quad y=10+3 t, \\quad z=10+3 t and L_2: x=-11+7 t \\quad y=-9+5 t \\quad z=-11+6 t intersect, are skew, or are parallel. If they intersect, determine the point of intersection; if not leave the remaining answer blanks empty.\nDo/are the lines: [ANS] Point of intersection: ([ANS], [ANS], [ANS])", "answer_v1": [ "intersect", "3", "1", "1" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV" ], "options_v1": [ [ "intersect", "skew", "parallel" ], [], [], [] ], "problem_v2": "Determine whether the lines L_1: x=4+3 t, \\quad y=26+7 t, \\quad z=5+3 t and L_2: x=-13+4 t \\quad y=-13+9 t \\quad z=-16+6 t intersect, are skew, or are parallel. If they intersect, determine the point of intersection; if not leave the remaining answer blanks empty.\nDo/are the lines: [ANS] Point of intersection: ([ANS], [ANS], [ANS])", "answer_v2": [ "intersect", "-5", "5", "-4" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV" ], "options_v2": [ [ "intersect", "skew", "parallel" ], [], [], [] ], "problem_v3": "Determine whether the lines L_1: x=10+4 t, \\quad y=7+2 t, \\quad z=7+3 t and L_2: x=-12+5 t \\quad y=-7+4 t \\quad z=-14+6 t intersect, are skew, or are parallel. If they intersect, determine the point of intersection; if not leave the remaining answer blanks empty.\nDo/are the lines: [ANS] Point of intersection: ([ANS], [ANS], [ANS])", "answer_v3": [ "intersect", "-2", "1", "-2" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV" ], "options_v3": [ [ "intersect", "skew", "parallel" ], [], [], [] ] }, { "id": "Geometry_0135", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "calculus", "line", "plane", "Multivariable", "Geometry", "Distance", "Point", "Line", "Dot Product" ], "problem_v1": "Find the distance from the point (4, 3, 4) to the line $x=0, \\quad y=3+4 t,\\quad z=4+2 t.$ [ANS]", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance from the point (1, 5, 1) to the line $x=0, \\quad y=5+2 t,\\quad z=1+5 t.$ [ANS]", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance from the point (2, 4, 2) to the line $x=0, \\quad y=4+3 t,\\quad z=2+2 t.$ [ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0136", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [ "symmetric equations for lines in space" ], "problem_v1": "Consider the line with these parametric equations: $\\quad \\quad x=-21 t+5$ $\\quad \\quad y=-9 t+2$ $\\quad \\quad z=3 t+2$. One set of symmetric equations for this line is\n\\frac{x-5}{-21}=\\frac{y-y_0}{b}=\\frac{z-z_0}{c}, where $y_0=$ [ANS], $z_0=$ [ANS], $b=$ [ANS], and $c=$ [ANS]. Another set of symmetric equations for this line is\n\\frac{x}{7}=\\frac{y-y_1}{b}=\\frac{z-z_1}{c}, where $y_1=$ [ANS], $z_1=$ [ANS], $b=$ [ANS], and $c=$ [ANS].", "answer_v1": [ "2", "2", "-9", "3", "-0.142857142857143", "2.71428571428571", "3", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the line with these parametric equations: $\\quad \\quad x=-2 t-8$ $\\quad \\quad y=-6 t+8$ $\\quad \\quad z=-12 t-7$. One set of symmetric equations for this line is\n\\frac{x+8}{-2}=\\frac{y-y_0}{b}=\\frac{z-z_0}{c}, where $y_0=$ [ANS], $z_0=$ [ANS], $b=$ [ANS], and $c=$ [ANS]. Another set of symmetric equations for this line is\n\\frac{x}{1}=\\frac{y-y_1}{b}=\\frac{z-z_1}{c}, where $y_1=$ [ANS], $z_1=$ [ANS], $b=$ [ANS], and $c=$ [ANS].", "answer_v2": [ "8", "-7", "-6", "-12", "32", "41", "3", "6" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the line with these parametric equations: $\\quad \\quad x=-16 t-4$ $\\quad \\quad y=-6 t+2$ $\\quad \\quad z=12 t-4$. One set of symmetric equations for this line is\n\\frac{x+4}{-16}=\\frac{y-y_0}{b}=\\frac{z-z_0}{c}, where $y_0=$ [ANS], $z_0=$ [ANS], $b=$ [ANS], and $c=$ [ANS]. Another set of symmetric equations for this line is\n\\frac{x}{8}=\\frac{y-y_1}{b}=\\frac{z-z_1}{c}, where $y_1=$ [ANS], $z_1=$ [ANS], $b=$ [ANS], and $c=$ [ANS].", "answer_v3": [ "2", "-4", "-6", "12", "3.5", "-7", "3", "-6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0137", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines", "level": "3", "keywords": [], "problem_v1": "Give a parametric equation of the line which passes through $A(3,1,1)$ and $B(5,-1,-1)$. Use $t$ as the parameter for all of your answers.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v1": [ "3+2*t; 1+(-2)*t; 1+(-2)*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Give a parametric equation of the line which passes through $A(-5,5,-4)$ and $B(-7,10,-6)$. Use $t$ as the parameter for all of your answers.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v2": [ "-5+(-2)*t; 5+5*t; -4+(-2)*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Give a parametric equation of the line which passes through $A(-2,1,-2)$ and $B(-1,-2,-4)$. Use $t$ as the parameter for all of your answers.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v3": [ "-2+1*t; 1+(-3)*t; -2+(-2)*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0138", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "Multivariable", "Geometry", "Vector" ], "problem_v1": "Find the equation of the plane through the point $(2,5,7)$ that is parallel to the line $\\mathbf{r}=(3\\mathbf{i}+2\\mathbf{j}-2\\mathbf{k})+t(\\mathbf{i}+2\\mathbf{j}+7 \\mathbf{k})$ and perpendicular to the plane $4x+5y+6z=14$. Write the equation in the form indicated. Equation: [ANS] $(x-2)+$ [ANS] $(y-$ [ANS]) $-\\,3(z-$ [ANS]) $=0$", "answer_v1": [ "-23", "22", "5", "7" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the equation of the plane through the point $(2,5,7)$ that is parallel to the line $\\mathbf{r}=(3\\mathbf{i}+2\\mathbf{j}-2\\mathbf{k})+t(\\mathbf{i}+2\\mathbf{j}+1 \\mathbf{k})$ and perpendicular to the plane $4x+5y+6z=14$. Write the equation in the form indicated. Equation: [ANS] $(x-2)+$ [ANS] $(y-$ [ANS]) $-\\,3(z-$ [ANS]) $=0$", "answer_v2": [ "7", "-2", "5", "7" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the equation of the plane through the point $(2,5,7)$ that is parallel to the line $\\mathbf{r}=(3\\mathbf{i}+2\\mathbf{j}-2\\mathbf{k})+t(\\mathbf{i}+2\\mathbf{j}+3 \\mathbf{k})$ and perpendicular to the plane $4x+5y+6z=14$. Write the equation in the form indicated. Equation: [ANS] $(x-2)+$ [ANS] $(y-$ [ANS]) $-\\,3(z-$ [ANS]) $=0$", "answer_v3": [ "-3", "6", "5", "7" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Geometry_0139", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find an equation of the plane through the three points given: $P=\\left(0,0,4\\right), Q=\\left(1,2,2\\right), R=\\left(-2,1,5\\right)$ [ANS] $=20$", "answer_v1": [ "4*x+3*y+5*z" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the plane through the three points given: $P=\\left(0,1,0\\right), Q=\\left(-3,0,4\\right), R=\\left(-2,-2,-2\\right)$ [ANS] $=-14$", "answer_v2": [ "14*x+(-14)*y+7*z" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the plane through the three points given: $P=\\left(0,0,2\\right), Q=\\left(-2,-3,1\\right), R=\\left(3,4,5\\right)$ [ANS] $=2$", "answer_v3": [ "-5*x+3*y+1*z" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0140", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the equation of the plane which passes through $O$ and is parallel to $3x+y+z=3$ [ANS] $x+$ [ANS] $y+z=$ [ANS]", "answer_v1": [ "3", "1", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the equation of the plane which passes through $O$ and is parallel to $6y-5x+z=-7$ [ANS] $x+$ [ANS] $y+z=$ [ANS]", "answer_v2": [ "-5", "6", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the equation of the plane which passes through $O$ and is parallel to $y-2x+z=-5$ [ANS] $x+$ [ANS] $y+z=$ [ANS]", "answer_v3": [ "-2", "1", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0141", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find a vector $\\mathbf{n}$ normal to the plane with the equation $3x+y+2z=5$. $\\mathbf{n}=\\left< 3,\\right.$ [ANS], [ANS] $\\left.\\right>$", "answer_v1": [ "1", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find a vector $\\mathbf{n}$ normal to the plane with the equation $6y-5x-5z=-3$. $\\mathbf{n}=\\left<-5,\\right.$ [ANS], [ANS] $\\left.\\right>$", "answer_v2": [ "6", "-5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find a vector $\\mathbf{n}$ normal to the plane with the equation $y-2x-3z=1$. $\\mathbf{n}=\\left<-2,\\right.$ [ANS], [ANS] $\\left.\\right>$", "answer_v3": [ "1", "-3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0142", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Write an equation of the plane with normal vector $\\bf{n}$ $=\\left<5,2,2\\right>$ passing through the point $\\left(4,-4,-3 \\right)$ in scalar form [ANS] $=24$", "answer_v1": [ "20*x+8*y+8*z" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write an equation of the plane with normal vector $\\bf{n}$ $=\\left<-8,8,-7\\right>$ passing through the point $\\left(-3, 8,-3 \\right)$ in scalar form [ANS] $=218$", "answer_v2": [ "16*y-16*x-14*z" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write an equation of the plane with normal vector $\\bf{n}$ $=\\left<-4,2,-4\\right>$ passing through the point $\\left(1,-6,-3 \\right)$ in scalar form [ANS] $=-20$", "answer_v3": [ "10*y-20*x-20*z" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0143", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "Angle", "Plane", "Multivariable", "Geometry" ], "problem_v1": "Find the angle in radians between the planes $-1x+z=1$ and $-1 y+z=1.$ [ANS]", "answer_v1": [ "1.0471975511966" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the angle in radians between the planes $5x+z=1$ and $-3 y+z=1.$ [ANS]", "answer_v2": [ "1.50873913581348" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the angle in radians between the planes $3x+z=1$ and $3 y+z=1.$ [ANS]", "answer_v3": [ "1.47062890563333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0144", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "Multivariable", "Geometry", "Distance", "Point", "Plane", "Dot Product" ], "problem_v1": "Find the distance from the point (3, 1, 1) to the plane $2\\!x-2\\!y-2\\!z=1.$ [ANS]", "answer_v1": [ "0.288675134594813" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance from the point (-5, 5,-4) to the plane $-2\\!x+5\\!y-2\\!z=-3.$ [ANS]", "answer_v2": [ "8.0075721739621" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance from the point (-2, 1,-2) to the plane $1\\!x-3\\!y-2\\!z=3.$ [ANS]", "answer_v3": [ "1.0690449676497" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0145", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "vectors", "dot product" ], "problem_v1": "(a) Find a vector $\\vec n$ perpendicular to the plane z=7x+6 y. $\\vec n=$ [ANS]\n(b) Find a vector $\\vec v$ parallel to the plane. $\\vec v=$ [ANS]", "answer_v1": [ "7i+6j-k", "6i-7j" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find a vector $\\vec n$ perpendicular to the plane z=2x+8 y. $\\vec n=$ [ANS]\n(b) Find a vector $\\vec v$ parallel to the plane. $\\vec v=$ [ANS]", "answer_v2": [ "2i+8j-k", "8i-2j" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find a vector $\\vec n$ perpendicular to the plane z=4x+6 y. $\\vec n=$ [ANS]\n(b) Find a vector $\\vec v$ parallel to the plane. $\\vec v=$ [ANS]", "answer_v3": [ "4i+6j-k", "6i-4j" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0146", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "cross product", "vectors" ], "problem_v1": "Find an equation for the plane through the points $(4,4,4), (-1,-1,0), (-1,0,1).$ The plane is [ANS]", "answer_v1": [ "5*y-x-5*z = -4" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find an equation for the plane through the points $(5,2,3), (2,0,-1), (0,-2,-1).$ The plane is [ANS]", "answer_v2": [ "8*y-8*x+2*z = -18" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find an equation for the plane through the points $(4,3,4), (-1,0,-1), (0,2,2).$ The plane is [ANS]", "answer_v3": [ "x+10*y-7*z = 6" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Geometry_0147", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "Implicit", "Plane" ], "problem_v1": "(A) If the positive z-axis points upward, an equation for a horizontal plane through the point $\\left(3,1,1\\right)$ is [ANS].\n(B) An equation for the plane perpendicular to the x-axis and passing through the point $\\left(3,1,1\\right)$ is [ANS].\n(C) An equation for the plane parallel to the xz-plane and passing through the point $\\left(3,1,1\\right)$ is [ANS].", "answer_v1": [ "z = 1", "x = 3", "y = 1" ], "answer_type_v1": [ "EQ", "EQ", "EQ" ], "options_v1": [ [], [], [] ], "problem_v2": "(A) If the positive z-axis points upward, an equation for a horizontal plane through the point $\\left(-5,5,-4\\right)$ is [ANS].\n(B) An equation for the plane perpendicular to the x-axis and passing through the point $\\left(-5,5,-4\\right)$ is [ANS].\n(C) An equation for the plane parallel to the xz-plane and passing through the point $\\left(-5,5,-4\\right)$ is [ANS].", "answer_v2": [ "z = -4", "x = -5", "y = 5" ], "answer_type_v2": [ "EQ", "EQ", "EQ" ], "options_v2": [ [], [], [] ], "problem_v3": "(A) If the positive z-axis points upward, an equation for a horizontal plane through the point $\\left(-2,1,-2\\right)$ is [ANS].\n(B) An equation for the plane perpendicular to the x-axis and passing through the point $\\left(-2,1,-2\\right)$ is [ANS].\n(C) An equation for the plane parallel to the xz-plane and passing through the point $\\left(-2,1,-2\\right)$ is [ANS].", "answer_v3": [ "z = -2", "x = -2", "y = 1" ], "answer_type_v3": [ "EQ", "EQ", "EQ" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0148", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "2", "keywords": [ "Linear function" ], "problem_v1": "Find the equation of the linear function $z=c+m x+n y$ whose graph contains the points $\\left(0,0,1\\right)$, $\\left(0,3,2\\right)$, and $\\left(1,0,3\\right)$. [ANS]", "answer_v1": [ "6*x+y-3*z = -3" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the linear function $z=c+m x+n y$ whose graph contains the points $\\left(0,0,5\\right)$, $\\left(0,-5,10\\right)$, and $\\left(-4,0,3\\right)$. [ANS]", "answer_v2": [ "10*x-20*y-20*z = -100" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the linear function $z=c+m x+n y$ whose graph contains the points $\\left(0,0,-3\\right)$, $\\left(0,-2,-2\\right)$, and $\\left(-2,0,-2\\right)$. [ANS]", "answer_v3": [ "2*x+2*y+4*z = -12" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Geometry_0149", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Planes", "level": "3", "keywords": [ "angle' 'intersection' 'plane" ], "problem_v1": "Find the angle between the plane $3x+y+z=1$ and the plane $2x-2 y-2 z=1$. $\\theta=$ [ANS] degrees. Notes: The angle between two planes is always less than or equal to $90^\\circ$.", "answer_v1": [ "79.9750121379243" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the angle between the plane $-5x+5 y-4 z=-3$ and the plane $-2x+5 y-2 z=-2$. $\\theta=$ [ANS] degrees. Notes: The angle between two planes is always less than or equal to $90^\\circ$.", "answer_v2": [ "22.8710998805387" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the angle between the plane $-2x+y-2 z=3$ and the plane $x-3 y-2 z=5$. $\\theta=$ [ANS] degrees. Notes: The angle between two planes is always less than or equal to $90^\\circ$.", "answer_v3": [ "84.8889103047113" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0150", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines with planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the intersection of the line and plane: $2x+y+z=-10, \\qquad \\mathbf{r}(t)=\\left<2,-1,-1\\right>+t\\left<-2,1,-1\\right>$ $P=($ [ANS], [ANS], [ANS] $)$", "answer_v1": [ "-4", "2", "-4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the intersection of the line and plane: $4y-4x-3z=3, \\qquad \\mathbf{r}(t)=\\left<-1,3,-1\\right>+t\\left<-2,-1,0\\right>$ $P=($ [ANS], [ANS], [ANS] $)$", "answer_v2": [ "7", "7", "-1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the intersection of the line and plane: $y-2x-2z=21, \\qquad \\mathbf{r}(t)=\\left<0,-2,-1\\right>+t\\left<2,3,3\\right>$ $P=($ [ANS], [ANS], [ANS] $)$", "answer_v3": [ "-6", "-11", "-10" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0152", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines with planes", "level": "5", "keywords": [ "Multivariable", "trigonometry" ], "problem_v1": "The axis of a light in a lighthouse is tilted, but the light shines perpendicular to its axis. When the light points east, it is inclined upward at 8 degrees. When it points north, it is inclined upward at 6 degrees. What is its maximum angle of elevation?\nAnswer=[ANS] degrees", "answer_v1": [ "9.95378266302744" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The axis of a light in a lighthouse is tilted, but the light shines perpendicular to its axis. When the light points east, it is inclined upward at 1 degree. When it points north, it is inclined upward at 10 degrees. What is its maximum angle of elevation?\nAnswer=[ANS] degrees", "answer_v2": [ "10.0478845635737" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The axis of a light in a lighthouse is tilted, but the light shines perpendicular to its axis. When the light points east, it is inclined upward at 4 degrees. When it points north, it is inclined upward at 7 degrees. What is its maximum angle of elevation?\nAnswer=[ANS] degrees", "answer_v3": [ "8.04267333913164" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0153", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines with planes", "level": "3", "keywords": [ "Intersect", "Unit", "Parallel", "Vector Equation", "Line", "Plane" ], "problem_v1": "Consider the planes $4\\!x+3\\!y+4\\!z=1$ and $4\\!x+4\\!z=0.$ (A) Find the unique point P on the y-axis which is on both planes. ([ANS], [ANS], [ANS]) (B) Find a unit vector $\\bf{u}$ with positive first coordinate that is parallel to both planes. [ANS] $\\mathbf{i}$+[ANS] $\\mathbf{j}$+[ANS] $\\mathbf{k}$ (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes, $\\bf{r(t)=}$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v1": [ "0", "0.333333333333333", "0", "0.707106781186547", "0", "-0.707106781186547", "t*4/sqrt( 4**2 + 4**2 )", "1/3", "t*(-4)/sqrt( 4**2 + 4**2 )" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the planes $1\\!x+5\\!y+1\\!z=1$ and $1\\!x+1\\!z=0.$ (A) Find the unique point P on the y-axis which is on both planes. ([ANS], [ANS], [ANS]) (B) Find a unit vector $\\bf{u}$ with positive first coordinate that is parallel to both planes. [ANS] $\\mathbf{i}$+[ANS] $\\mathbf{j}$+[ANS] $\\mathbf{k}$ (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes, $\\bf{r(t)=}$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v2": [ "0", "0.2", "0", "0.707106781186547", "0", "-0.707106781186547", "t*1/sqrt( 1**2 + 1**2 )", "1/5", "t*(-1)/sqrt( 1**2 + 1**2 )" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the planes $2\\!x+4\\!y+2\\!z=1$ and $2\\!x+2\\!z=0.$ (A) Find the unique point P on the y-axis which is on both planes. ([ANS], [ANS], [ANS]) (B) Find a unit vector $\\bf{u}$ with positive first coordinate that is parallel to both planes. [ANS] $\\mathbf{i}$+[ANS] $\\mathbf{j}$+[ANS] $\\mathbf{k}$ (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes, $\\bf{r(t)=}$ [ANS] $\\bf{i}$+[ANS] $\\bf{j}$+[ANS] $\\bf{k}$", "answer_v3": [ "0", "0.25", "0", "0.707106781186547", "0", "-0.707106781186547", "t*2/sqrt( 2**2 + 2**2 )", "1/4", "t*(-2)/sqrt( 2**2 + 2**2 )" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Geometry_0154", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines with planes", "level": "3", "keywords": [ "Parametric", "Multivariable", "Geometry", "Line", "Parametric", "Perpendicular", "Plane", "Intersect" ], "problem_v1": "(A) Find the parametric equations for the line through the point P=(3, 1, 1) that is perpendicular to the plane $2x-2 y-2 z=1.$ Use \"t\" as your variable, t=0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x=[ANS]\ny=[ANS]\nz=[ANS]\n(B) At what point Q does this line intersect the yz-plane? Q=([ANS], [ANS], [ANS])", "answer_v1": [ "3 + t * 2", "1 + t * -2", "1 + t * -2", "0", "4", "4" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "(A) Find the parametric equations for the line through the point P=(-5, 5,-4) that is perpendicular to the plane $-2x+5 y-2 z=1.$ Use \"t\" as your variable, t=0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x=[ANS]\ny=[ANS]\nz=[ANS]\n(B) At what point Q does this line intersect the yz-plane? Q=([ANS], [ANS], [ANS])", "answer_v2": [ "-5 + t * -2", "5 + t * 5", "-4 + t * -2", "0", "-7.5", "1" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "(A) Find the parametric equations for the line through the point P=(-2, 1,-2) that is perpendicular to the plane $1x-3 y-2 z=1.$ Use \"t\" as your variable, t=0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x=[ANS]\ny=[ANS]\nz=[ANS]\n(B) At what point Q does this line intersect the yz-plane? Q=([ANS], [ANS], [ANS])", "answer_v3": [ "-2 + t * 1", "1 + t * -3", "-2 + t * -2", "0", "-5", "-6" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Geometry_0155", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Lines with planes", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "Find an equation for the plane containing the line in the $xy$-plane where $y=5$, and the line in the $x z$-plane where $z=7.5$. equation: [ANS]", "answer_v1": [ "z = -1.5*y+7.5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation for the plane containing the line in the $xy$-plane where $x=1$, and the line in the $y z$-plane where $z=2$. equation: [ANS]", "answer_v2": [ "z = -2*x+2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation for the plane containing the line in the $xy$-plane where $x=2$, and the line in the $y z$-plane where $z=3$. equation: [ANS]", "answer_v3": [ "z = -1.5*x+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0157", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Convert the following point from rectangular to cylindrical coordinates: (4\\sqrt{2},-4\\sqrt{2},5). $(r,\\theta,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v1": [ "(8,7*pi/4,5)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Convert the following point from rectangular to cylindrical coordinates: (1,1\\sqrt{3},-3). $(r,\\theta,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v2": [ "(2,1*pi/3,-3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Convert the following point from rectangular to cylindrical coordinates: (-2\\sqrt{2},2\\sqrt{2},1). $(r,\\theta,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v3": [ "(4,3*pi/4,1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0158", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Consider a rectangular coordinate system with origin at the center of the earth, $z$-axis through the North Pole, and $x$-axis through the prime-meridian. Find the rectangular coordinates of Paris, France ($48^{\\circ}48' \\text{N}$, $2^{\\circ}20' \\text{E}$). A minute is $1/60^{\\circ}$. Assume the earth is a sphere of radius $R=6367 \\text{km}$. Paris has coordinates [ANS]. Usage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v1": [ "(4190.4,170.746,4790.62)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Consider a rectangular coordinate system with origin at the center of the earth, $z$-axis through the North Pole, and $x$-axis through the prime-meridian. Find the rectangular coordinates of Sydney, Australia ($34^{\\circ} \\text{S}$, $151^{\\circ} \\text{E}$). A minute is $1/60^{\\circ}$. Assume the earth is a sphere of radius $R=6367 \\text{km}$. Sydney has coordinates [ANS]. Usage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v2": [ "(-4616.67,2559.04,-3560.39)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Consider a rectangular coordinate system with origin at the center of the earth, $z$-axis through the North Pole, and $x$-axis through the prime-meridian. Find the rectangular coordinates of Bangkok, Thailand ($13^{\\circ}45' \\text{N}$, $100^{\\circ}30' \\text{E}$). A minute is $1/60^{\\circ}$. Assume the earth is a sphere of radius $R=6367 \\text{km}$. Bangkok has coordinates [ANS]. Usage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v3": [ "(-1127.07,6080.96,1513.37)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0159", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Convert the following point from cylindrical to rectangular coordinates: \\left(6, \\frac{3 \\pi}{2}, 2.5\\right). $(x,y,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v1": [ "(0,-6,2.5)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Convert the following point from cylindrical to rectangular coordinates: \\left(10, \\frac{\\pi}{4},-7\\right). $(x,y,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v2": [ "(7.07107,7.07107,-7)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Convert the following point from cylindrical to rectangular coordinates: \\left(6, \\frac{2 \\pi}{3},-4.5\\right). $(x,y,z)=$ [ANS]\nUsage: To enter a point, for example $(x,y,z)$, type \"(x, y, z)\".", "answer_v3": [ "(-3,5.19615,-4.5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Geometry_0160", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "2", "keywords": [ "Coordinate", "Spherical", "Rectangular", "coordinates" ], "problem_v1": "What are the rectangular coordinates of the point whose cylindrical coordinates are $(r=8,\\ \\theta=3.8,\\ z=2)$? x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v1": [ "-6.32774169531533", "-4.89486312754175", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "What are the rectangular coordinates of the point whose cylindrical coordinates are $(r=0,\\ \\theta=0.8,\\ z=9)$? x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v2": [ "0", "0", "9" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "What are the rectangular coordinates of the point whose cylindrical coordinates are $(r=3,\\ \\theta=1.6,\\ z=2)$? x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v3": [ "-0.0875985669038664", "2.99872080912452", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0161", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "Coordinate", "Rectangular", "Spherical", "coordinates" ], "problem_v1": "What are the spherical coordinates of the point whose rectangular coordinates are $(4,\\ 3,\\ 1)$?\n$\\rho$=[ANS]\n$\\theta$=[ANS]\n$\\phi$=[ANS]", "answer_v1": [ "5.09901951359278", "0.643501108793284", "1.37340076694502" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "What are the spherical coordinates of the point whose rectangular coordinates are $(1,\\ 5,\\-4)$?\n$\\rho$=[ANS]\n$\\theta$=[ANS]\n$\\phi$=[ANS]", "answer_v2": [ "6.48074069840786", "1.37340076694502", "2.23599238180772" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "What are the spherical coordinates of the point whose rectangular coordinates are $(2,\\ 4,\\-2)$?\n$\\rho$=[ANS]\n$\\theta$=[ANS]\n$\\phi$=[ANS]", "answer_v3": [ "4.89897948556636", "1.10714871779409", "1.99133066207886" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Geometry_0162", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "4", "keywords": [ "Coordinate", "Cylindrical", "Spherical", "Surface", "coordinates", "cylindrical", "spherical" ], "problem_v1": "Match the given equation with the verbal description of the surface:\nA. Plane B. Elliptic or Circular Paraboloid C. Circular Cylinder D. Half plane E. Cone F. Sphere [ANS] 1. $r=2\\cos(\\theta)$ [ANS] 2. $\\rho \\cos(\\phi)=4$ [ANS] 3. $\\theta=\\frac{\\pi}{3}$ [ANS] 4. $\\rho=4$ [ANS] 5. $\\phi=\\frac{\\pi}{3}$ [ANS] 6. $r^2+z^2=16$ [ANS] 7. $\\rho=2\\cos(\\phi)$ [ANS] 8. $r=4$ [ANS] 9. $z=r^2$", "answer_v1": [ "C", "A", "D", "F", "E", "F", "F", "C", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Match the given equation with the verbal description of the surface:\nA. Half plane B. Plane C. Elliptic or Circular Paraboloid D. Cone E. Circular Cylinder F. Sphere [ANS] 1. $\\rho=4$ [ANS] 2. $\\rho=2\\cos(\\phi)$ [ANS] 3. $r=2\\cos(\\theta)$ [ANS] 4. $r^2+z^2=16$ [ANS] 5. $\\theta=\\frac{\\pi}{3}$ [ANS] 6. $r=4$ [ANS] 7. $\\phi=\\frac{\\pi}{3}$ [ANS] 8. $\\rho \\cos(\\phi)=4$ [ANS] 9. $z=r^2$", "answer_v2": [ "F", "F", "E", "F", "A", "E", "D", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Match the given equation with the verbal description of the surface:\nA. Plane B. Half plane C. Cone D. Elliptic or Circular Paraboloid E. Sphere F. Circular Cylinder [ANS] 1. $\\rho=4$ [ANS] 2. $r=2\\cos(\\theta)$ [ANS] 3. $\\theta=\\frac{\\pi}{3}$ [ANS] 4. $z=r^2$ [ANS] 5. $r^2+z^2=16$ [ANS] 6. $\\rho=2\\cos(\\phi)$ [ANS] 7. $r=4$ [ANS] 8. $\\phi=\\frac{\\pi}{3}$ [ANS] 9. $\\rho \\cos(\\phi)=4$", "answer_v3": [ "E", "F", "B", "D", "E", "E", "F", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Geometry_0163", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "Distance", "Point", "Plane", "vector" ], "problem_v1": "What is the distance from the point (8, 6, 3) to the xz-plane? Distance=[ANS]", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the distance from the point (0, 10,-7) to the xz-plane? Distance=[ANS]", "answer_v2": [ "10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the distance from the point (3, 7,-5) to the xz-plane? Distance=[ANS]", "answer_v3": [ "7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Geometry_0164", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "coordinates", "cylindrical", "spherical", "integrals", "calculus" ], "problem_v1": "Find an equation for the paraboloid $z=4-(x^2+y^2)$ in cylindrical coordinates. (Type theta for $\\theta$ in your answer.) equation: [ANS]", "answer_v1": [ "z = 4-r^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation for the plane $y=1$ in cylindrical coordinates. (Type theta for $\\theta$ in your answer.) equation: [ANS]", "answer_v2": [ "r = 1/[sin(theta)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation for the paraboloid $z=2-(x^2+y^2)$ in cylindrical coordinates. (Type theta for $\\theta$ in your answer.) equation: [ANS]", "answer_v3": [ "z = 2-r^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Geometry_0165", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "vectors" ], "problem_v1": "(a) What is the distance from an arbitrary point $(x,y)$ in $\\mathbb{R}^2$ to the $x$-axis? [ANS]\n(b) Find a fully simplified equation for the set of all points $P=(x,y)$ equidistant from the $x$-axis and the point $\\left(3,1\\right)$. Plot a graph of this set of points and describe this set geometrically. $y=$ [ANS]", "answer_v1": [ "y", "[(x-3)^2+1]/2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) What is the distance from an arbitrary point $(x,y)$ in $\\mathbb{R}^2$ to the $x$-axis? [ANS]\n(b) Find a fully simplified equation for the set of all points $P=(x,y)$ equidistant from the $x$-axis and the point $\\left(-5,5\\right)$. Plot a graph of this set of points and describe this set geometrically. $y=$ [ANS]", "answer_v2": [ "y", "([x-(-5)]^2+25)/10" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) What is the distance from an arbitrary point $(x,y)$ in $\\mathbb{R}^2$ to the $x$-axis? [ANS]\n(b) Find a fully simplified equation for the set of all points $P=(x,y)$ equidistant from the $x$-axis and the point $\\left(-2,1\\right)$. Plot a graph of this set of points and describe this set geometrically. $y=$ [ANS]", "answer_v3": [ "y", "([x-(-2)]^2+1)/2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Geometry_0166", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "Implicit", "Plane", "Distance", "Point" ], "problem_v1": "You are given the following points: $A=(11,-8,-19)$, $B=(-18, 0, 2)$, $C=(-8,-5, 17)$.\nWhich point is closest to the yz-plane? [ANS] What is the distance from the yz-plane to this point? [ANS]\nWhich point is farthest from the xy-plane? [ANS] What is the distance from the xy-plane to this point? [ANS]\nWhich point lies on the xz-plane? [ANS]", "answer_v1": [ "C", "8", "A", "19", "B" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ], [], [ "A", "B", "C" ] ], "problem_v2": "You are given the following points: $A=(-8, 18, 16)$, $B=(-16, 0,-4)$, $C=(10, 3,-17)$.\nWhich point is closest to the yz-plane? [ANS] What is the distance from the yz-plane to this point? [ANS]\nWhich point is farthest from the xy-plane? [ANS] What is the distance from the xy-plane to this point? [ANS]\nWhich point lies on the xz-plane? [ANS]", "answer_v2": [ "A", "8", "C", "17", "B" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ], [], [ "A", "B", "C" ] ], "problem_v3": "You are given the following points: $A=(-5,-12, 17)$, $B=(-20, 0, 9)$, $C=(8, 15,-18)$.\nWhich point is closest to the yz-plane? [ANS] What is the distance from the yz-plane to this point? [ANS]\nWhich point is farthest from the xy-plane? [ANS] What is the distance from the xy-plane to this point? [ANS]\nWhich point lies on the xz-plane? [ANS]", "answer_v3": [ "A", "5", "C", "18", "B" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ], [], [ "A", "B", "C" ] ] }, { "id": "Geometry_0167", "subject": "Geometry", "topic": "Vector geometry", "subtopic": "Coordinate systems", "level": "3", "keywords": [ "homogeneous coordinates" ], "problem_v1": "WARNING: Because this problem is/contains multiple choice, you are permitted A LIMITED NUMBER OF ATTEMPTS. Use them wisely. The column vector $\\left[\\begin{array}{c} 5 \\\\ 2 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c} 3 \\\\ 5 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c}-4 \\\\-4 \\\\ 1 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS]. The column vector $\\left[\\begin{array}{c} 2 \\\\-3 \\\\ 3 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS].", "answer_v1": [ "point", "5", "2", "vector", "3", "5", "point", "-4", "-4", "1", "vector", "2", "-3", "3" ], "answer_type_v1": [ "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS", "NV", "NV", "NV" ], "options_v1": [ [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [], [ "point", "vector" ], [], [], [] ], "problem_v2": "WARNING: Because this problem is/contains multiple choice, you are permitted A LIMITED NUMBER OF ATTEMPTS. Use them wisely. The column vector $\\left[\\begin{array}{c}-9 \\\\ 9 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c}-7 \\\\-3 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c} 9 \\\\-4 \\\\-7 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS]. The column vector $\\left[\\begin{array}{c}-4 \\\\ 1 \\\\-9 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS].", "answer_v2": [ "point", "-9", "9", "vector", "-7", "-3", "point", "9", "-4", "-7", "vector", "-4", "1", "-9" ], "answer_type_v2": [ "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS", "NV", "NV", "NV" ], "options_v2": [ [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [], [ "point", "vector" ], [], [], [] ], "problem_v3": "WARNING: Because this problem is/contains multiple choice, you are permitted A LIMITED NUMBER OF ATTEMPTS. Use them wisely. The column vector $\\left[\\begin{array}{c}-4 \\\\ 2 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c}-5 \\\\ 1 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS] and [ANS]. The column vector $\\left[\\begin{array}{c}-6 \\\\-3 \\\\ 6 \\\\ 1 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS]. The column vector $\\left[\\begin{array}{c} 9 \\\\ 8 \\\\-6 \\\\ 0 \\end{array}\\right]$ is in homogeneous coordinates. What is it in standard notation? Answer: It is a [ANS] with coordinates [ANS], [ANS], and [ANS].", "answer_v3": [ "point", "-4", "2", "vector", "-5", "1", "point", "-6", "-3", "6", "vector", "9", "8", "-6" ], "answer_type_v3": [ "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS", "NV", "NV", "NV" ], "options_v3": [ [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [ "point", "vector" ], [], [], [], [ "point", "vector" ], [], [], [] ] } ]