Question 1

Find a set of parametric equations for the line that passes through the given points. Show all your work.
(9,8, -2) ,(-3, -6,4)

Accepted Answer

Answers may vary. One possible answer is shown below.
$$x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t$$

Question 2

Find the angle between the two planes in degrees. Round to a tenth of a degree.
$$\begin{array} { l } 
2 x - y - 4 z = 4 \
- 3 x - 2 y + 4 z = 4
\end{array}$$

Accepted Answer

A)  $11.5 ^ { \circ }$ 
B)  $54.1 ^ { \circ }$ 
C)  $144.1 ^ { \circ }$ 
D)  $39.0 ^ { \circ }$ 
E)  $2.5 ^ { \circ }$ 
A)  $11.5 ^ { \circ }$ 
B)  $54.1 ^ { \circ }$ 
C)  $144.1 ^ { \circ }$ 
D)  $39.0 ^ { \circ }$ 
E)  $2.5 ^ { \circ }$ C

Question 3

Find the area of the triangle with the given vertices.
$$( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$$

Accepted Answer

A)  0 
B)  1 
C)  $\frac { 7 \sqrt { 11 } } { 2 }$ 
D)  $\frac { 7 \sqrt { 11 } } { 4 }$ 
E)  $7 \sqrt { 11 }$ 
A)  0 
B)  1 
C)  $\frac { 7 \sqrt { 11 } } { 2 }$ 
D)  $\frac { 7 \sqrt { 11 } } { 4 }$ 
E)  $7 \sqrt { 11 }$ C

Question 4

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,8, -2) ,(-3, -6,4)

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(Essay)

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Question 1

Correct Answer:

Verified

Answers may vary. One possible answer is shown below.
$x = 9 - 12 t , y = 8 - 14 t , z = - 2 + 6 t$

Accepted Answer

Answers may vary. On

Question 5

Find the torque on the crankshaft $\mathbf { V }$ using the data shown in the figure. Round to the nearest tenth of a foot-pound.

$\begin{array}{l}
\|\mathbf{V}\|=1.6 \mathrm{ft} \
\|\mathbf{F}\|=20 \mathrm{lb} \
\theta=60^{\circ}
\end{array}$

Accepted Answer

A)  $27.7 \mathrm { ft } - \mathrm { lb }$ 
B)  $16.0 \mathrm { ft } - \mathrm { lb }$ 
C)  $55.4 \mathrm { ft } - \mathrm { lb }$ 
D)  $32.0 \mathrm { ft } - \mathrm { lb }$ 
E)  0 
A)  $27.7 \mathrm { ft } - \mathrm { lb }$ 
B)  $16.0 \mathrm { ft } - \mathrm { lb }$ 
C)  $55.4 \mathrm { ft } - \mathrm { lb }$ 
D)  $32.0 \mathrm { ft } - \mathrm { lb }$ 
E)  0

Question 6

Find the center and radius of the sphere.
$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$$

Accepted Answer

A)  center: $( - 8,3,3 )$; radius: 8 
B)  center: $( 8 , - 3 , - 3 )$; radius: 8 
C)  center: $( 8,3,3 )$; radius: 8 
D)  center: $( - 8,3,3 )$; radius: 64 
E)  center: $( 8 , - 3 , - 3 )$; radius: 64 
A)  center: $( - 8,3,3 )$; radius: 8 
B)  center: $( 8 , - 3 , - 3 )$; radius: 8 
C)  center: $( 8,3,3 )$; radius: 8 
D)  center: $( - 8,3,3 )$; radius: 64 
E)  center: $( 8 , - 3 , - 3 )$; radius: 64

Question 7

Find the area of the parallelogram that has the vectors as adjacent sides.
$$\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangle$$

Accepted Answer

A)  12 
B)  $\sqrt { 41 }$ 
C)  $2 \sqrt { 2 }$ 
D)  15 
E)  $5 \sqrt { 41 }$ 
A)  12 
B)  $\sqrt { 41 }$ 
C)  $2 \sqrt { 2 }$ 
D)  15 
E)  $5 \sqrt { 41 }$

Question 8

Find the volume of the parallelpiped with the given vertices. $$\begin{array} { l } 
A ( 8 , - 5 , - 4 ) , B ( 2,2,5 ) , C ( 9,4 , - 12 ) , D ( 3,11 , - 3 ) \
E ( 8 , - 6 , - 10 ) , F ( 2,1 , - 1 ) , G ( 9,3 , - 18 ) , H ( 3,10 , - 9 )
\end{array}$$ )

Accepted Answer

A)  369 
B)  405 
C)  321 
D)  455 
E)  430 
A)  369 
B)  405 
C)  321 
D)  455 
E)  430

Question 9

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle - 7,1 , - 8 \rangle$$

Accepted Answer

A)  - 14 
B)  16 
C)  $\sqrt { 14 }$ 
D)  $2 \sqrt { 114 }$ 
E)  $\sqrt { 114 }$ 
A)  - 14 
B)  16 
C)  $\sqrt { 14 }$ 
D)  $2 \sqrt { 114 }$ 
E)  $\sqrt { 114 }$

Question 10

Find the vector $\mathbf { z }$, given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$, and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$. $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

Accepted Answer

A)  $\langle - 10,6,14 \rangle$ 
B)  $\langle - 3,5 , - 7 \rangle$ 
C)  $\langle - 5,3 , - 7 \rangle$ 
D)  $\langle 5 , - 3 , - 7 \rangle$ 
E)  $\langle 6 , - 3 , - 6 \rangle$ 
A)  $\langle - 10,6,14 \rangle$ 
B)  $\langle - 3,5 , - 7 \rangle$ 
C)  $\langle - 5,3 , - 7 \rangle$ 
D)  $\langle 5 , - 3 , - 7 \rangle$ 
E)  $\langle 6 , - 3 , - 6 \rangle$

Question 11

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

Accepted Answer

Let point @#LAT-DLM&, point @#LAT-DLM&, and point @#LAT-DLM&.
\[\begin

Question 12

Write the component form of the vector described below.
Initial point: $( - 7,2 , - 1 )$
Terminal point: $( - 5 , - 9 , - 6 )$

Accepted Answer

A)  $\langle 2 , - 11 , - 5 \rangle$ 
B)  $\langle - 2,11,5 \rangle$ 
C)  $\langle - 5 , - 9 , - 6 \rangle$ 
D)  $\langle 2 , - 11 , - 7 \rangle$ 
E)  $\langle 35 , - 18,6 \rangle$ 
A)  $\langle 2 , - 11 , - 5 \rangle$ 
B)  $\langle - 2,11,5 \rangle$ 
C)  $\langle - 5 , - 9 , - 6 \rangle$ 
D)  $\langle 2 , - 11 , - 7 \rangle$ 
E)  $\langle 35 , - 18,6 \rangle$

Question 13

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle 0 , - 4 , - 4 \rangle$$

Accepted Answer

A)  $- 8$ 
B)  8 
C)  $2 \sqrt { 2 }$ 
D)  $8 \sqrt { 2 }$ 
E)  $4 \sqrt { 2 }$ 
A)  $- 8$ 
B)  8 
C)  $2 \sqrt { 2 }$ 
D)  $8 \sqrt { 2 }$ 
E)  $4 \sqrt { 2 }$

Question 14

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$.
$$\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$$

Accepted Answer

A)  $- 93$ 
B)  4 
C)  $- 18 \mathbf { i } - 30 \mathbf { j } - 45 \mathbf { k }$ 
D)  $- 3$ 
E)  $7 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$ 
A)  $- 93$ 
B)  4 
C)  $- 18 \mathbf { i } - 30 \mathbf { j } - 45 \mathbf { k }$ 
D)  $- 3$ 
E)  $7 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$

Question 15

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors
$$\mathbf { u } = \langle - 3 , - 5 , - 8 \rangle , \mathbf { v } = \langle - 1 , - 6,3 \rangle , \mathbf { w } = \langle - 8 , - 7,6 \rangle$$

Accepted Answer

A)  $- 378$ 
B)  463 
C)  $- 463$ 
D)  283 
E)  0 
A)  $- 378$ 
B)  463 
C)  $- 463$ 
D)  283 
E)  0

Question 16

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point $( - 3,6 , - 2 )$ and is parallel to the $y z$-plane

Accepted Answer

A)  $y = 6$ 
B)  $z = - 2$ 
C)  $x + y + z = 1$ 
D)  $y + z = 4$ 
E)  $x = - 3$ 
A)  $y = 6$ 
B)  $z = - 2$ 
C)  $x + y + z = 1$ 
D)  $y + z = 4$ 
E)  $x = - 3$

Question 17

Find the coordinates of the point located three units in front of the $y z$-plane, eight units to the right of the $x z$-plane, and five units above the $x y$-plane.

Accepted Answer

A)  $( 3,5,8 )$ 
B)  $( 3,8,5 )$ 
C)  $( 3 , - 8,5 )$ 
D)  $( 3,5 , - 8 )$ 
E)  $( 3,11,16 )$ 
A)  $( 3,5,8 )$ 
B)  $( 3,8,5 )$ 
C)  $( 3 , - 8,5 )$ 
D)  $( 3,5 , - 8 )$ 
E)  $( 3,11,16 )$

Question 18

Find the coordinates of the point located four units in front of the $y z$-plane, nine units to the right of the $x z$-plane, and three units below the $x y$-plane.

Accepted Answer

A)  $( 4 , - 3,9 )$ 
B)  $( 4,9 , - 3 )$ 
C)  $( 4 , - 9 , - 3 )$ 
D)  $( 4 , - 3 , - 9 )$ 
E)  $( 4,13,10 )$ 
A)  $( 4 , - 3,9 )$ 
B)  $( 4,9 , - 3 )$ 
C)  $( 4 , - 9 , - 3 )$ 
D)  $( 4 , - 3 , - 9 )$ 
E)  $( 4,13,10 )$

Question 19

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. 
$x=-8+7 t$
$(-5,-3,3) \text {, parallel to } y=7+3 t$
$z=-9-5 t$

Accepted Answer

Answers may vary. On

Question 20

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.
$x = - 1 + 5 t$
$( 6 , - 8,9 ) , \quad y = - 3 + t$
$z = - 2 - 6 t$

Accepted Answer

A)  $5 x + y - 6 z = 0$ 
B)  $5 x + y - 6 z - 32 = 0$ 
C)  $5 x + y - 6 z + 32 = 0$ 
D)  $6 x - 8 y + 9 z - 32 = 0$ 
E)  $6 x - 8 y + 9 z + 32 = 0$ 
A)  $5 x + y - 6 z = 0$ 
B)  $5 x + y - 6 z - 32 = 0$ 
C)  $5 x + y - 6 z + 32 = 0$ 
D)  $6 x - 8 y + 9 z - 32 = 0$ 
E)  $6 x - 8 y + 9 z + 32 = 0$

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,8, -2) ,(-3, -6,4)

Find the angle between the two planes in degrees. Round to a tenth of a degree. 2x-y-4z=4 -3x-2y+4z=4

Find the area of the triangle with the given vertices. $( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangle$

Find the volume of the parallelpiped with the given vertices. A(8,-5,-4),B(2,2,5),C(9,4,-12),D(3,11,-3) E(8,-6,-10),F(2,1,-1),G(9,3,-18),H(3,10,-9) )

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$ , and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$ . $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 4 , - 4 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = \langle - 3 , - 5 , - 8 \rangle , \mathbf { v } = \langle - 1 , - 6,3 \rangle , \mathbf { w } = \langle - 8 , - 7,6 \rangle$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point $( - 3,6 , - 2 )$ and is parallel to the $y z$ -plane

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Find the coordinates of the point located four units in front of the $y z$ -plane, nine units to the right of the $x z$ -plane, and three units below the $x y$ -plane.

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x=-8+7 t$ $(-5,-3,3) \text {, parallel to } y=7+3 t$ $z=-9-5 t$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = - 1 + 5 t$ $( 6 , - 8,9 ) , \quad y = - 3 + t$ $z = - 2 - 6 t$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Exam 10: Analytic Geometry in Three Dimensions

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,8, -2) ,(-3, -6,4)

Find the angle between the two planes in degrees. Round to a tenth of a degree. 2x-y-4z=4 -3x-2y+4z=4

Find the area of the triangle with the given vertices. (−4,−2,2),(−9,−7,0),(−4,−3,5)( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )(−4,−2,2),(−9,−7,0),(−4,−3,5)

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,92,72),(32,12,9)\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)(9,29​,27​),(23​,21​,9)

Find the torque on the crankshaft V\mathbf { V }V using the data shown in the figure. Round to the nearest tenth of a foot-pound. \|\|=1.6 \|\|=20 \theta=6

Find the center and radius of the sphere. x2+y2+z2−16x+6y+6z+18=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0x2+y2+z2−16x+6y+6z+18=0

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangleu=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩

Find the volume of the parallelpiped with the given vertices. A(8,-5,-4),B(2,2,5),C(9,4,-12),D(3,11,-3) E(8,-6,-10),F(2,1,-1),G(9,3,-18),H(3,10,-9) )

Find the magnitude of the vector v\mathbf { v }v . v=⟨−7,1,−8⟩\mathbf { v } = \langle - 7,1 , - 8 \ranglev=⟨−7,1,−8⟩

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (−9,6,−1),(−8,3,−2),(3,6,0)( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )(−9,6,−1),(−8,3,−2),(3,6,0) )

Write the component form of the vector described below. Initial point: (−7,2,−1)( - 7,2 , - 1 )(−7,2,−1) Terminal point: (−5,−9,−6)( - 5 , - 9 , - 6 )(−5,−9,−6)

Find the magnitude of the vector v\mathbf { v }v . v=⟨0,−4,−4⟩\mathbf { v } = \langle 0 , - 4 , - 4 \ranglev=⟨0,−4,−4⟩

Find the dot product of u\mathbf { u }u and v\mathbf { v }v . u=9i−5j−9k,v=−2i+6j+5k\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }u=9i−5j−9k,v=−2i+6j+5k

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (−3,6,−2)( - 3,6 , - 2 )(−3,6,−2) and is parallel to the yzy zyz -plane

Find the coordinates of the point located three units in front of the yzy zyz -plane, eight units to the right of the xzx zxz -plane, and five units above the xyx yxy -plane.

Find the coordinates of the point located four units in front of the yzy zyz -plane, nine units to the right of the xzx zxz -plane, and three units below the xyx yxy -plane.

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. x=−8+7tx=-8+7 tx=−8+7t (−5,−3,3), parallel to y=7+3t(-5,-3,3) \text {, parallel to } y=7+3 t(−5,−3,3), parallel to y=7+3t z=−9−5tz=-9-5 tz=−9−5t

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. x=−1+5tx = - 1 + 5 tx=−1+5t (6,−8,9),y=−3+t( 6 , - 8,9 ) , \quad y = - 3 + t(6,−8,9),y=−3+t z=−2−6tz = - 2 - 6 tz=−2−6t

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Find the area of the triangle with the given vertices. $( - 4 , - 2,2 ) , ( - 9 , - 7,0 ) , ( - 4 , - 3,5 )$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 9 , \frac { 9 } { 2 } , \frac { 7 } { 2 } \right) , \left( \frac { 3 } { 2 } , \frac { 1 } { 2 } , 9 \right)$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangle$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$ )

Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 4 , - 4 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point $( - 3,6 , - 2 )$ and is parallel to the $y z$ -plane

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Find the coordinates of the point located four units in front of the $y z$ -plane, nine units to the right of the $x z$ -plane, and three units below the $x y$ -plane.

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x=-8+7 t$ $(-5,-3,3) \text {, parallel to } y=7+3 t$ $z=-9-5 t$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = - 1 + 5 t$ $( 6 , - 8,9 ) , \quad y = - 3 + t$ $z = - 2 - 6 t$