Exam 12: Vectors and the Geometry of Space

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons. Select the correct answer.

(Multiple Choice)

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Question 121

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

(Multiple Choice)

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Question 122

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

(Short Answer)

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Question 123

$\text { Find an equation of the plane through the origin and parallel to the plane } - 5 x + 2 y - 5 z = - 1$

(Short Answer)

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Question 124

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

(Multiple Choice)

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Question 125

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$ -axis.

(Multiple Choice)

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Question 126

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

(Multiple Choice)

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Question 127

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 4 z = 50$

(Short Answer)

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Question 128

Find the midpoint of the line segment joining the given points. Select the correct answer. $( 0 , - 1 , - 3 )$ and $( 2 , - 3 , - 1 )$

(Multiple Choice)

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Question 129

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 5 \mathbf { j }$ and $\mathbf { b } = - 5 \mathbf { j } + 2 \mathbf { k }$ I. $\mathbf { a } \cdot \mathbf { b }$ ㅍ. $\mathbf { a } \times \mathbf { b }$ III. The angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree)

(Short Answer)

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Question 130

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

(Multiple Choice)

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Question 131

$\text { Find that the midpoint of the line segment from } P 1 ( - 14,14,9 ) \text { to } P 2 ( 2,8,3 ) \text {. }$

(Short Answer)

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Question 132

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$ Select the correct answer.

(Multiple Choice)

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Question 133

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

(Multiple Choice)

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Question 134

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

(Short Answer)

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Question 135

$\text { Find the unit vectors that are parallel to the tangent line to the curve } y = 2 x ^ { 2 } \text { at the point } ( 2,8 ) \text {. }$

(Short Answer)

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Question 136

Find the values of $x$ such that the vectors $( 2 x , x , 9 )$ and $( 6 , x , 3 )$ are orthogonal. Select the correct answer.

(Multiple Choice)

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Question 137

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ . Select the correct answer.

(Multiple Choice)

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Question 138

$\text { Find } | \mathbf { u } \times \mathbf { v } | \text { correct to three decimal places where } | \mathbf { u } | = 14 , | \mathbf { v } | = 9 , \angle \theta = 80 ^ { \circ } \text {. }$

(Short Answer)

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Question 139

$\text { Find an equation of the sphere with center } ( 6 , - 3,6 ) \text { that touches the } x y \text {-plane. }$

(Short Answer)

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Question 140

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons. Select the correct answer.

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

$\text { Find an equation of the plane through the origin and parallel to the plane } - 5 x + 2 y - 5 z = - 1$

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$ -axis.

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 4 z = 50$

Find the midpoint of the line segment joining the given points. Select the correct answer. $( 0 , - 1 , - 3 )$ and $( 2 , - 3 , - 1 )$

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

$\text { Find that the midpoint of the line segment from } P 1 ( - 14,14,9 ) \text { to } P 2 ( 2,8,3 ) \text {. }$

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$ Select the correct answer.

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

$\text { Find the unit vectors that are parallel to the tangent line to the curve } y = 2 x ^ { 2 } \text { at the point } ( 2,8 ) \text {. }$

Find the values of $x$ such that the vectors $( 2 x , x , 9 )$ and $( 6 , x , 3 )$ are orthogonal. Select the correct answer.

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ . Select the correct answer.

$\text { Find } | \mathbf { u } \times \mathbf { v } | \text { correct to three decimal places where } | \mathbf { u } | = 14 , | \mathbf { v } | = 9 , \angle \theta = 80 ^ { \circ } \text {. }$

$\text { Find an equation of the sphere with center } ( 6 , - 3,6 ) \text { that touches the } x y \text {-plane. }$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

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Partial Derivatives

Multiple Integrals

Vector Calculus

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Exam 12: Vectors and the Geometry of Space

Find the length of each side of the triangle ABCA B CABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. A(−1,0,1),B(1,1,−1),C(1,1,1)A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )A(−1,0,1),B(1,1,−1),C(1,1,1)

Find the length of the median of side ABA BAB of the triangle with vertices A(−8,6,−9),B(−6,−6,−5)A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )A(−8,6,−9),B(−6,−6,−5) and C(8,−6,2)C ( 8 , - 6,2 )C(8,−6,2) .

Find an equation of the plane through the origin and parallel to the plane −5x+2y−5z=−1\text { Find an equation of the plane through the origin and parallel to the plane } - 5 x + 2 y - 5 z = - 1 Find an equation of the plane through the origin and parallel to the plane −5x+2y−5z=−1

Find the distance from (−10,2,−8)( - 10,2 , - 8 )(−10,2,−8) to the xyx yxy -planes.

Find an equation for the surface obtained by rotating the parabola y=3x2y = 3 x ^ { 2 }y=3x2 about the yyy -axis.

Find the length of each side of the triangle ABCA B CABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer. A(−1,0,1),B(1,1,−1),C(1,1,1)A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )A(−1,0,1),B(1,1,−1),C(1,1,1)

Find the distance (correct to two decimal places) between the given parallel planes. 6x+7y−2z=12 and 14x+16y−4z=506 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 4 z = 506x+7y−2z=12 and 14x+16y−4z=50

Find the midpoint of the line segment joining the given points. Select the correct answer. (0,−1,−3)( 0 , - 1 , - 3 )(0,−1,−3) and (2,−3,−1)( 2 , - 3 , - 1 )(2,−3,−1)

Find the unit vectors that are parallel to the tangent line to the curve y=2x2y = 2 x ^ { 2 }y=2x2 at the point (2,8)( 2,8 )(2,8) .

Find that the midpoint of the line segment from P1(−14,14,9) to P2(2,8,3). \text { Find that the midpoint of the line segment from } P 1 ( - 14,14,9 ) \text { to } P 2 ( 2,8,3 ) \text {. } Find that the midpoint of the line segment from P1(−14,14,9) to P2(2,8,3).

Find the point at which the line given by the parametric equations below intersects the plane. 2x+4y−3z=−48;x=10+7t,y=−10,z=7t2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t2x+4y−3z=−48;x=10+7t,y=−10,z=7t Select the correct answer.

Find an equation for the surface consisting of all points PPP for which the distance from PPP to the xxx -axis is two times the distance from PPP to the yzy zyz -plane.

Find the distance between the planes. 5x−2y+z−4=0,5x−2y+z+6=05 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 05x−2y+z−4=0,5x−2y+z+6=0

Find the values of xxx such that the vectors (2x,x,9)( 2 x , x , 9 )(2x,x,9) and (6,x,3)( 6 , x , 3 )(6,x,3) are orthogonal. Select the correct answer.

Find an equation for the surface consisting of all points that are equidistant from the point (0,−2,0)( 0 , - 2,0 )(0,−2,0) and the plane y=2y = 2y=2 . Select the correct answer.

Find an equation of the sphere with center (6,−3,6) that touches the xy-plane. \text { Find an equation of the sphere with center } ( 6 , - 3,6 ) \text { that touches the } x y \text {-plane. } Find an equation of the sphere with center (6,−3,6) that touches the xy-plane.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the length of the median of side $A B$ of the triangle with vertices $A ( - 8,6 , - 9 ) , B ( - 6 , - 6 , - 5 )$ and $C ( 8 , - 6,2 )$ .

$\text { Find an equation of the plane through the origin and parallel to the plane } - 5 x + 2 y - 5 z = - 1$

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$ -axis.

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 4 z = 50$

Find the midpoint of the line segment joining the given points. Select the correct answer. $( 0 , - 1 , - 3 )$ and $( 2 , - 3 , - 1 )$

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

$\text { Find that the midpoint of the line segment from } P 1 ( - 14,14,9 ) \text { to } P 2 ( 2,8,3 ) \text {. }$

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$ Select the correct answer.

Find an equation for the surface consisting of all points $P$ for which the distance from $P$ to the $x$ -axis is two times the distance from $P$ to the $y z$ -plane.

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

Find the values of $x$ such that the vectors $( 2 x , x , 9 )$ and $( 6 , x , 3 )$ are orthogonal. Select the correct answer.

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ . Select the correct answer.

$\text { Find an equation of the sphere with center } ( 6 , - 3,6 ) \text { that touches the } x y \text {-plane. }$