Deck 11: Analytic Geometry In Three Dimensions

Question 1

Find a set of symmetric equations of the line that passes through the given points. $( 5,0,5 ) , ( 1,6 , - 3 )$

A) $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z - 5 } { - 8 }$
B) $\frac { x - 5 } { - 4 } = \frac { y } { - 6 } = \frac { z - 5 } { - 8 }$
C) $\frac { x - 5 } { - 4 } = \frac { y } { - 8 } = \frac { z - 5 } { 6 }$
D) $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$
E) $\frac { x + 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$

Answer 1

$\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z - 5 } { - 8 }$

Question 2

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: $( 2 , - 5,3 )$ Parallel to: $v = ( 4 , - 3 , - 6 )$

A)Parametric equations: $x = 4 + 3 t , y = - 3 - 5 t , z = - 6 + 2 t$
B)Parametric equations: $x = - 6 + 2 t , y = 4 - 5 t , z = - 3 + 3 t$
C)Parametric equations: $x = - 6 + 2 t , y = 4 - 5 t , z = - 6 + 3 t$
D)Parametric equations: $x = 2 + 4 t , y = - 5 - 3 t , z = 3 - 6 t$
E)Parametric equations: $x = - 6 + 3 t , y = - 3 - 5 t , z = 4 + 2 t$

Answer 2

Parametric equations: $x = 2 + 4 t , y = - 5 - 3 t , z = 3 - 6 t$

Question 3

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: $( 3 , - 6,2 )$ Parallel to: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

A)Parametric equations: $x = 3 + t , y = - 6 - 4 t , z = 2 + 2 t$
B)Parametric equations: $x = 2 + 2 t , y = 2 - 4 t , z = 2 + t$
C)Parametric equations: $x = 2 + 2 t , y = 3 - 4 t , z = - 6 + t$
D)Parametric equations: $x = - 6 + 2 t , y = - 6 - 4 t , z = 3 + t$
E)Parametric equations: $x = 3 + 2 t , y = - 6 - 4 t , z = 2 + t$

Answer 3

Parametric equations: $x = 3 + 2 t , y = - 6 - 4 t , z = 2 + t$

Question 4

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: $( - 6,2,0 )$ Parallel to: $\mathrm { v } = \frac { 3 } { 7 } \mathrm { i } + \frac { - 3 } { 7 } \mathrm { j } - 5 \mathrm { k }$

A)Parametric equations: $x = - 6 - 3 t , y = 4 - 3 t , z = - 5 t$
B)Parametric equations: $x = - 2 + 3 t , y = 5 - 3 t , z = - 5 t$
C)Parametric equations: $x = - 5 t , y = 5 - 3 t , z = - 5 t$
D)Parametric equations: $x = - 6 + 3 t , y = 2 - 3 t , z = - 35 t$
E)Parametric equations: $x = 3 t , y = - 5 t , z = - 3 t$

Question 5

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: (5,0,2)
Parallel to: $x = 3 + 3 t , y = 5 - 2 t , z = - 7 + t$

A)Symmetric equations: $\frac { x + 3 } { 3 } = \frac { y - 2 } { - 2 } = z - 2$
B)Symmetric equations: $\frac { x - 2 } { 3 } = \frac { y - 2 } { - 2 } = z - 2$
C)Symmetric equations: $\frac { x - 5 } { - 2 } = \frac { y } { 3 } = z - 2$
D)Symmetric equations: $\frac { z - 5 } { 3 } = \frac { y } { - 2 } = x - 2$
E)Symmetric equations: $\frac { x - 5 } { 3 } = \frac { y } { - 2 } = z - 2$

Question 6

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: $( 3,2,6 )$ Perpendicular to: $n = i$

A) $- 3 x = 0$
B) $x = 0$
C) $x - 3 = 0$
D) $3 x = 0$
E) $x + 3 = 0$

Question 7

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: $( 6 , - 2,2 )$ Parallel to: $\mathbf { v } = ( 3 , - 4 , - 10 )$

A)Symmetric equations: $\frac { x - 6 } { 3 } = \frac { y + 2 } { - 4 } = \frac { z - 2 } { - 10 }$
B)Symmetric equations: $\frac { x - 6 } { - 4 } = \frac { y + 2 } { - 4 } = \frac { z - 2 } { - 10 }$
C)Symmetric equations: $\frac { x - 6 } { 3 } = \frac { y + 2 } { - 10 } = \frac { z - 2 } { 3 }$
D)Symmetric equations: $\frac { x - 6 } { - 4 } = \frac { y + 2 } { 3 } = \frac { z - 2 } { - 10 }$
E)Symmetric equations: $\frac { x - 6 } { - 10 } = \frac { y + 2 } { 3 } = \frac { z - 2 } { - 4 }$

Question 8

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: $( - 7,0,2 )$ Parallel to: $\mathbf { v } = 4 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$

A)Symmetric equations: $\frac { x + 7 } { - 3 } = \frac { y } { 5 } = \frac { z - 2 } { 4 }$
B)Symmetric equations: $\frac { x + 7 } { 4 } = \frac { y } { 5 } = \frac { z - 2 } { - 3 }$
C)Symmetric equations: $\frac { x + 7 } { 4 } = \frac { z } { 5 } = \frac { y - 2 } { - 3 }$
D)Symmetric equations: $\frac { x + 5 } { 4 } = \frac { y } { 5 } = \frac { z - 2 } { - 3 }$
E)Symmetric equations: $\frac { x + 7 } { 4 } = \frac { y + 7 } { 5 } = \frac { z - 2 } { - 3 }$

Question 9

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: $( 0,0,8 )$ Perpendicular to: $x = 1 - t , y = 2 + t , z = 4 - 4 t$

A) $x - y + 4 z - 32 = 0$
B) $x + y - 4 z - 32 = 0$
C) $- x - y - 4 z + 32 = 0$
D) $- x + y + 4 z + 32 = 0$
E) $- x + y - 4 z + 32 = 0$

Question 10

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: $( 2,0 , - 7 )$ Perpendicular to: $\mathbf { n } = - 7 \mathbf { k }$

A) $z + 7 = 0$
B) $7 z = 0$
C) $z - 7 = 0$
D) $z = 0$
E) $- 7 z = 0$

Question 11

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: (0,0,0)
Perpendicular to: $\mathbf { n } = - 2 \mathbf { j } + 6 \mathbf { k }$

A) $x + 2 y + 6 z = 0$
B) $6 y + 2 z = 0$
C) $x - 2 y + 6 z = 0$
D) $- 2 y + 6 z = 0$
E) $- 2 y - 6 z = 0$

Question 12

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: (0,0,0)
Parallel to: $\mathbf { v } = ( 5,6,7 )$

A)Parametric equations: $x = 6 t , y = 5 t , z = 7 t$
B)Parametric equations: $x = 5 t , y = 6 t , z = 7 t$
C)Parametric equations: $x = 6 t , y = 7 t , z = 5 t$
D)Parametric equations: $x = 7 t , y = 6 t , z = 5 t$
E)Parametric equations: $x = 7 t , y = 6 t , z = 7 t$

Question 13

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: $( 4 , - 4,5 )$ Parallel to: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

A)Symmetric equations: $\frac { x - 4 } { 2 } = \frac { y - 4 } { - 4 } = z - 5$
B)Symmetric equations: $\frac { x - 4 } { - 4 } = \frac { y + 4 } { 2 } = z - 5$
C)Symmetric equations: $\frac { x - 4 } { 2 } = \frac { y - 4 } { - 4 } = z - 4$
D)Symmetric equations: $\frac { x - 5 } { 2 } = \frac { y + 4 } { - 4 } = z - 4$
E)Symmetric equations: $\frac { x - 4 } { 2 } = \frac { y + 4 } { - 4 } = z - 5$

Question 14

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: $( - 2,2,0 )$ Parallel to: $\mathbf { v } = \frac { 3 } { 7 } \mathrm { i } - \frac { 4 } { 7 } \mathbf { j } - 3 \mathrm { k }$

A)Symmetric equations: $\frac { x } { 3 } = \frac { y - 2 } { - 21 } = \frac { z } { - 4 }$
B)Symmetric equations: $\frac { x + 2 } { 3 } = \frac { y - 2 } { 3 } = \frac { z } { - 21 }$
C)Symmetric equations: $\frac { x + 2 } { 3 } = \frac { y - 2 } { - 4 } = \frac { z - 6 } { - 3 }$
D)Symmetric equations: $\frac { x + 2 } { 3 } = \frac { y - 2 } { - 4 } = \frac { z } { - 21 }$
E)Symmetric equations: $\frac { x + 2 } { - 4 } = \frac { y } { 3 } = \frac { z } { - 21 }$

Question 15

Find a set of parametric equations of the line that passes through the given points. $( 5,0,6 ) , ( 1,6 , - 3 )$

A) $x = 5 - 4 t , y = 6 t , z = - 9 t$
B) $x = 5 - 4 t , y = 5 + 6 t , z = 6 - 9 t$
C) $x = 5 - 4 t , y = 6 t , z = 6 - 9 t$
D) $x = 6 - 4 t , y = 1 - 6 t , z = 6 - 9 t$
E) $x = 6 - 4 t , y = 6 t , z = 5 - 9 t$

Question 16

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: $( 2,0,2 )$ Parallel to: $x = 3 + 4 t , y = 5 - 5 t , z = - 7 + 3 t$

A)Parametric equations: $x = 4 t , y = - 5 t , z = 2 + 3 t$
B)Parametric equations: $x = 2 + 4 t , y = 2 - 5 t , z = 2 + 3 t$
C)Parametric equations: $x = 2 + 4 t , y = - 5 t , z = 2 + 3 t$
D)Parametric equations: $x = 2 + 4 t , y = 2 - 5 t , z = 2 - 2 t$
E)Parametric equations: $x = 2 + - 5 t , y = 3 t , z = 2 + 4 t$

Question 17

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: $( 7,8,2 )$ Perpendicular to: $\mathbf { n } = - 4 \mathbf { i } + \mathbf { j } - 5 \mathbf { k }$

A) $- 4 x + - 4 y - 5 z + 30 = 0$
B) $- 4 y - 5 z + 30 = 0$
C) $- 4 x + y 30 z + - 5 = 0$
D) $- 4 x + y - 5 z + 30 = 0$
E) $- 4 x + y - 5 z - 30 = 0$

Question 18

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. )
Point: $( - 7,2,2 )$ Parallel to: $\mathbf { v } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$

A)Parametric equations: $x = 3 + - 7 t , y = 2 + 6 t , z = 2 - 2 t$
B)Parametric equations: $x = - 7 + 3 t , y = 6 + 2 t , z = 2 - 2 t$
C)Parametric equations: $x = - 7 + 2 t , y = 2 + 6 t , z = 2 - 2 t$
D)Parametric equations: $x = - 7 + 3 t , y = 2 + 6 t , z = 2 - 2 t$
E)Parametric equations: $x = 2 + 3 t , y = 2 + 6 t , z = 2 - 2 t$

Question 19

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line.
Point: (7,0,0)
Parallel to: $\mathbf { v } = ( 5,6,7 )$

A)Symmetric equations: $\frac { x } { 5 } = \frac { y } { 7 } = \frac { z - 7 } { 7 }$
B)Symmetric equations: $\frac { x } { 7 } = \frac { y } { 6 } = \frac { z } { 5 }$
C)Symmetric equations: $\frac { x - 7 } { 5 } = \frac { y } { 6 } = \frac { z } { 7 }$
D)Symmetric equations: $\frac { x } { 5 } = \frac { y - 7 } { 5 } = \frac { z } { 7 }$
E)Symmetric equations: $\frac { x - 7 } { 6 } = \frac { y } { 5 } = \frac { z } { 7 }$

Question 20

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Point: $( 3,0,0 )$ Perpendicular to: $x = 3 - t , y = 2 - 6 t , z = 4 + t$

A) $- x - 6 y - z + 3 = 0$
B) $x - 6 y + z + 3 = 0$
C) $- x + 6 y + z + 3 = 0$
D) $- x - 6 y + z + 3 = 0$
E) $- x - 6 y - z - 3 = 0$

Question 21

Find a set of parametric equations of the line.
Passes through $( 5,6,6 )$ and is parallel to the xz-plane and the yz-plane

A) $\begin{array} { l } x = 5 + \mathrm { t } \\y = 6 \\z = 6\end{array}$
B) $\begin{array} { l } x = 5 \\y = 6 \\z = 6 + t\end{array}$
C) $\begin{array} { l } x = 5 \\y = 6 + t \\z = 6\end{array}$
D) $\begin{array} { l } x = 5 \\y = 6 + t \\z = 6 + t\end{array}$
E) $\begin{array} { l } x = 5 + t \\y = 6 + t \\z = 6\end{array}$

Question 22

Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,4,-1), (1,6,2), (-5,-3,6)

A)35x -38y -33z = 0
B)4x -4y 1z = 0
C)4x -4y 1z + 259 = 0
D)35x -38y -33z + 259 = 0
E)35x -38y -33z - 259 = 0

Question 23

Find a set of parametric equations of the line.
Passes through $( - 4,2,3 )$ and is parallel to the xy-plane and the yz-plane

A) $\begin{array} { l } x = - 4 \\y = 2 \\z = 3 + t\end{array}$
B) $\begin{array} { l } x = - 4 + t \\y = 2 \\z = 3\end{array}$
C) $\begin{array} { l } x = - 4 \\y = 2 + t \\z = 3 + t\end{array}$
D) $\begin{array} { l } x = - 4 \\y = 2 + t \\z = 3\end{array}$
E) $\begin{array} { l } x = - 4 \\y = 2 \\z = 3 - t\end{array}$

Question 24

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (-2,-3,-5)and is parallel to the yz-plane.

A)x + y + z = -10
B)y = -3
C)z = -5
D)y + z = -8
E)x = -2

Question 25

Find a set of parametric equations of the line.
Passes through $( 4,6,5 )$ and is perpendicular to $3 x + 6 y - z = 6$ .

A) $\begin{array} { l } x = 4 + 3 t \\y = 6 + 6 t \\z = 5 + t\end{array}$
B) $\begin{array} { l } x = 4 + 3 t \\y = 6 + 6 t \\z = - 5 - t\end{array}$
C) $\begin{array} { l } x = 4 - 3 t \\y = 6 - 6 t \\z = 5 - t\end{array}$
D) $\begin{array} { l } x = 4 + 3 t \\y = 6 + 6 t \\z = 5 - t\end{array}$
E) $\begin{array} { l } x = 4 + 3 t \\y = 6 - 6 t \\z = 5 - t\end{array}$

Question 26

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] $\begin{array} { l } ( - 7 , - 8 , - 1 ) \\x = 3 + 2 t \\y = - 3 - 2 t \\z = 2 + 4 t\end{array}$

A)7x + 8y + z + 2 = 0
B)7x + 8y + z - 2 = 0
C)x - y + 2z - 1 = 0
D)x - y + 2z + 1 = 0
E)x - y + 2z = 0

Question 27

Find a set of symmetric equations of the line that passes through the given points. $( - 5,8,12 ) , ( 1 , - 2,19 )$

A) $\frac { x + 5 } { 6 } = \frac { y - 8 } { 7 } = \frac { z - 12 } { 6 }$
B) $\frac { x + 5 } { 6 } = \frac { y - 8 } { - 10 } = \frac { z - 12 } { 7 }$
C) $\frac { x - 5 } { 6 } = \frac { y - 8 } { - 10 } = \frac { z - 12 } { 7 }$
D) $\frac { x + 5 } { 6 } = \frac { y + 8 } { - 10 } = \frac { z + 12 } { 7 }$
E) $\frac { x - 5 } { - 10 } = \frac { y + 8 } { - 10 } = \frac { z - 12 } { 7 }$

Question 28

Find a set of symmetric equations of the line that passes through the given points. $( 3,2,0 ) , ( 9,12,12 )$

A) $\frac { x + 3 } { 10 } = \frac { y - 2 } { 6 } = \frac { z } { 12 }$
B) $\frac { x + 3 } { 6 } = \frac { y + 2 } { 10 } = \frac { z } { 12 }$
C) $\frac { x - 3 } { 6 } = \frac { y + 2 } { 10 } = \frac { z } { 12 }$
D) $\frac { x - 3 } { 6 } = \frac { y - 2 } { 10 } = \frac { z } { 12 }$
E) $\frac { x - 3 } { 6 } = \frac { y - 2 } { 10 } = \frac { - z } { 12 }$

Question 29

Determine whether the planes are parallel,orthogonal,or neither.
x + 2y + z = 6
-2x - 4y - 2z = -10

A)orthogonal
B)parallel
C)neither

Question 30

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,5,6),n = 2i- 4j+ 4k

A)x - 2y + 2 z + 2 = 0
B)4x - 5y - 6z + 4 = 0
C)x - 2y + 2 z - 2 = 0
D)x - 2y + 2 z = 0
E)4x - 5y - 6z - 4 = 0

Question 31

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (3,-2,-7)and (-2,6,3)and is perpendicular to the plane -x - 4y - z = 4.

A)32x - 15y + 28z - 262 = 0
B)31x - 16y + 30z + 85 = 0
C)x + 4y + z = 0
D)32x + 15y + 28z - 130 = 0
E)x + 4y + z - 12 = 0

Question 32

Find a set of parametric equations of the line.
Passes through $( - 7,4,7 )$ and is perpendicular to $- x + 4 y + z = 5$ .

A) $\begin{array} { l } x = - 7 - 7 t \\y = 4 - 4 t \\z = 7 + t\end{array}$
B) $\begin{array} { l } x = - 7 - 7 t \\y = 4 - 4 t \\z = 7 - t\end{array}$
C) $\begin{array} { l } x = - 7 - 7 t \\y = 4 + 4 t \\z = 7 - t\end{array}$
D) $\begin{array} { l } x = - 7 - 7 t \\y = 4 + 4 t \\z = 7 + t\end{array}$
E) $\begin{array} { l } x = - 7 + 7 t \\y = 4 + 4 t \\z = 7 + t\end{array}$

Question 33

Determine whether the planes are parallel,orthogonal,or neither.
5x - 2y - 5z = -2
X - 3y + 6z = -6

A)neither
B)parallel
C)orthogonal

Question 34

Find a set of parametric equations of the line that passes through the given points. $( 2,3,0 ) , ( 9,9,12 )$

A) $x = 2 - 7 t , y = 3 + 6 t , z = 12 t$
B) $x = 2 + 7 t , y = 3 + 6 t , z = 12 t$
C) $x = 2 + 7 t , y = 3 - 6 t , z = - 12 t$
D) $x = 2 - 7 t , y = 3 - 6 t , z = - 12 t$
E) $x = 2 + 6 t , y = 3 + 7 t , z = 12 t$

Question 35

Find the angle of intersection of the planes in degrees.Round to a tenth of a degree.
3x + 2y - 6z = -5
-6x + 3y + z = 1

A)112.3°
B)2.0°
C)-22.3°
D)109.4°
E)20.8°

Question 36

Determine whether the planes are parallel,orthogonal,or neither. 5x - y + z = -4
-x - 6y - z = -2

A)neither
B)orthogonal
C)parallel

Question 37

Find a set of parametric equations of the line that passes through the given points. $( - 4,7,13 ) , ( 1 , - 4,17 )$

A) $x = - 4 + 5 t , y = 7 - 11 t , z = 13 - 4 t$
B) $x = - 4 - 5 t , y = 7 - 11 t , z = 13 - 4 t$
C) $x = - 4 + 5 t , y = - 11 t , z = 13 + 4 t$
D) $x = - 4 + 5 t , y = 7 - 11 t , z = 4 t$
E) $x = - 4 + 5 t , y = 7 - 11 t , z = 13 + 4 t$

Question 38

Find a set of parametric equations of the line.
Passes through $( 3 , - 4 , - 3 )$ and is parallel to $\mathbf { v } = ( 6 , - 6,3 )$ .

A) $\begin{array} { l } x = 3 + 6 t \\y = - 4 - 6 t \\z = - 3 + 3 t\end{array}$
B) $\begin{array} { l } x = 3 - 6 t \\y = - 4 - 6 t \\z = - 3 + 3 t\end{array}$
C) $\begin{array} { l } x = 3 + 6 t \\y = - 4 + 6 t \\z = - 3 + 3 t\end{array}$
D) $\begin{array} { l } x = 3 - 6 t \\y = - 4 - 6 t \\z = - 3 - 3 t\end{array}$
E) $\begin{array} { l } x = 3 + 6 t \\y = - 4 - 6 t \\z = - 3 - 3 t\end{array}$

Question 39

Find the general form of the equation of the plane with the given characteristics.
Passes through $( 7,4,6 )$ and is parallel to the yz-plane

A) $x = 0$
B) $- 7 x = 0$
C) $x + 7 = 0$
D) $7 x = 0$
E) $x - 7 = 0$

Question 40

Find the general form of the equation of the plane with the given characteristics.
Passes through $( 3,8,6 )$ and is parallel to the xz-plane

A) $y + 8 = 0$
B) $- 8 y = 0$
C) $y - 8 = 0$
D) $8 y = 0$
E) $y = 0$

Question 41

Find the angle between the two planes. $\begin{array} { l } x - 4 y + z = - 3 \\3 x + 3 z + 4 = 0\end{array}$

A)71.5 $^\circ$
B)72.5 $^\circ$
C)73.5 $^\circ$
D)70.5 $^\circ$
E)69.5 $^\circ$

Question 42

Determine whether the planes are parallel,orthogonal,or neither. $\begin{array} { l } x - 9 y - z = 5 \\4 x - 36 y - 4 z = - 2\end{array}$

A)Neither
B)Orthogonal
C)Parallel

Question 43

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (6,2,9),n = 8i + 6j + 9k

A)8x + 6y + 9 z = 0
B)8x + 6y + 9 z + 141 = 0
C)8x + 6y + 9 z - 141 = 0
D)6x + 2y + 9z + 141 = 0
E)6x + 2y + 9z - 141 = 0

Question 44

Determine whether the planes are parallel,orthogonal,or neither.
5x - 6y - 6z = 0
3x - 4y + 2z = -4

A)parallel
B)orthogonal
C)neither

Question 45

Find the angle of intersection of the planes in degrees.Round to a tenth of a degree. 5x + y - z = -3
-3x - y + z = -1

A)167.7°
B)-80.6°
C)170.6°
D)44.6°
E)3.0°

Question 46

Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (3,1,6), (-6,-1,6), (3,5,-6)

A)2x - 9y - 3z + 21 = 0
B)2x - 9y - 3z = 0
C)3x + y + 6z = 0
D)2x - 9y - 3z - 21 = 0
E)3x + y + 6z - 252 = 0

Question 47

Find a set of parametric equations of the line.
Passes through $( - 1,6 , - 3 )$ and is parallel to $\mathbf { v } = 5 \mathbf { i } - \mathbf { j }$ .

A) $\begin{array} { l } x = - 1 + 5 t \\y = 6 - t \\z = 3\end{array}$
B) $\begin{array} { l } x = - 1 + 5 t \\y = - 6 - t \\z = - 3\end{array}$
C) $\begin{array} { l } x = 1 + 5 t \\y = 6 - t \\z = 3\end{array}$
D) $\begin{array} { l } x = - 1 + 5 t \\y = 6 - t \\z = - 3\end{array}$
E) $\begin{array} { l } x = 1 + 5 t \\y = 6 - t \\z = - 3\end{array}$

Question 48

Find the general form of the equation of the plane passing through the three points. $( 0,0,0 ) , ( 5,6,7 ) , ( - 6,7,7 )$

A) $- 7 x + 71 y + 77 z = 0$
B) $- 7 x - 77 y - 71 z = 0$
C) $- 7 x - 77 y + 71 z = 0$
D) $- 7 x + 77 y - 71 z = 0$
E) $- 7 x + 77 y + 71 z = 0$

Question 49

Find the angle between the two planes in degrees.Round to a tenth of a degree.
3x - 4y + z = -6
2x + y + 3z = 0

A)15.2°
B)71.9°
C)14.7°
D)74.8°
E)1.3°

Question 50

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (5,-1,-4)and is parallel to the yz-plane.

A)y = -1
B)z = -4
C)x + y + z = 0
D)y + z = -5
E)x = 5

Question 51

Determine whether the planes are parallel,orthogonal,or neither.
5x - 4y + z = -2
3x + 4y + z = -5

A)parallel
B)orthogonal
C)neither

Question 52

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (5,5,-2)and (4,2,1)and is perpendicular to the plane -3x - 2y + 4z = 1.

A)6x + 5y + 7z - 41 = 0
B)3x + 2y - 4z + 33 = 0
C)6x - 5y + 7z - 9 = 0
D)3x + 2y - 4z = 0
E)7x + 6y + 5z - 55 = 0

Question 53

Determine whether the planes are parallel,orthogonal,or neither. $\begin{array} { l } 3 x - z = 3 \\7 x + y + 21 z = 7\end{array}$

A)Neither
B)Parallel
C)Orthogonal

Question 54

Find the angle,in degrees,between two adjacent sides of the pyramid shown below.Round to the nearest tenth of a degree.[Note: The base of the pyramid is not considered a side.] P(8,0,0),Q(8,8,0),R(0,8,0),S(4,4,4)

A)54.7°
B)30°
C)2.1°
D)90°
E)45°

Question 55

Find the angle between the two planes. $\begin{array} { l } 3 x - 4 y + 5 z = 6 \\x + y - z = 2\end{array}$

A)81.6°
B)83.2°
C)83.8°
D)80.6°
E)79.6°

Question 56

Determine whether the planes are parallel,orthogonal,or neither.
6x + y - z = 2
-18x - 3y + 3z = -4

A)parallel
B)orthogonal
C)neither

Question 57

Find the angle between the two planes in degrees.Round to a tenth of a degree.
5x - 6y - 4z = -5
X + y + 5z = -1

A)114.5°
B)27.4°
C)117.4°
D)24.7°
E)2.0°

Question 58

Find the angle between the two planes in degrees.Round to a tenth of a degree. $$\begin{array} { l } 3 x - 4 y + 1 z = - 6 \\2 x + 1 y - 3 z = 0\end{array}$$

A)88.0°
B)90.8°
C)10.8°
D)89.8°
E)89.7°

Question 59

Find the angle between the two planes. $\begin{array} { l } x + y - z = 3 \\4 x - 5 y - 4 z = 5\end{array}$

A)77.7°
B)76.7°
C)79.7°
D)78.7°
E)75.7°

Question 60

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] $\begin{array} { l } ( - 3,3 , - 7 ) \\x = 4 - t \\y = 1 - 3 t \\z = 3 - 5 t\end{array}$

A)x + 3y + 5z = 0
B)x + 3y + 5z - 29 = 0
C)x + 3y + 5z + 29 = 0
D)3x - 3y + 7z - 29 = 0
E)3x - 3y + 7z + 29 = 0

Question 61

Find u × v and show that it is orthogonal to both u and v.
$\begin{array} { l } \mathbf { u } = \frac { 5 } { 7 } \mathbf { i } \\\mathbf { v } = \frac { 6 } { 7 } \mathbf { j } - 49 \mathbf { k }\end{array}$

A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0\end{array}$
B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathrm { i } - \frac { 30 } { 49 } \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0\end{array}$
C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0\end{array}$
D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0\end{array}$
E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { j } + \frac { 30 } { 49 } \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0\end{array}$

Question 62

Find u × v and show that it is orthogonal to both u and v. $$\begin{array} { l } \mathbf { u } = 12 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } \\\mathbf { v } = \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }\end{array}$$

A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 73 \mathbf { i } - 41 \mathbf { j } - 41 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 73 \mathbf { i } - 41 \mathbf { j } + 73 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 73 \mathbf { i } - 41 \mathbf { j } + 54 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 41 \mathbf { i } + 73 \mathbf { j } + 54 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 41 \mathbf { i } + 73 \mathbf { j } - 41 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$

Question 63

Find symmetric equations for the line through the point and parallel to the specified vector.
(9,-6,-5),parallel to

Question 64

Find a set of parametric equations for the line through the point and parallel to the specified vector.

Question 65

Find symmetric equations for the line through the point and parallel to the specified line.

Question 66

Find a set of parametric equations for the line that passes through the given points.

Question 67

Use the vectors u and v to find u × v. $\mathbf { u } = 5 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }$ $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

A) $\mathbf { u } \times \mathbf { v } = 24 \mathbf { i } + 29 \mathbf { j } - 23 \mathbf { k }$
B) $\mathbf { u } \times \mathbf { v } = 29 \mathbf { i } + 24 \mathbf { j } - 23 \mathbf { k }$
C) $\mathbf { u } \times \mathbf { v } = 24 \mathbf { i } - 23 \mathbf { j } + 29 \mathbf { k }$
D) $\mathbf { u } \times \mathbf { v } = 29 \mathbf { i } - 23 \mathbf { j } + 24 \mathbf { k }$
E) $\mathbf { u } \times \mathbf { v } = - 23 \mathbf { i } + 29 \mathbf { j } + 24 \mathbf { k }$

Question 68

Use the vectors u and v to find v × u. $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

A) $\mathbf { v } \times \mathbf { u } = 11 \mathbf { i } + 7 \mathbf { j } - 8 \mathbf { k }$
B) $\mathbf { v } \times \mathbf { u } = 7 \mathbf { i } - 11 \mathbf { j } - 8 \mathbf { k }$
C) $\mathbf { v } \times \mathbf { u } = - 7 \mathbf { i } + 11 \mathbf { j } + 7 \mathbf { k }$
D) $\mathbf { v } \times \mathbf { u } = 11 \mathbf { i } + 11 \mathbf { j } + 8 \mathbf { k }$
E) $\mathbf { v } \times \mathbf { u } = 7 \mathbf { i } - 8 \mathbf { j } + 8 \mathbf { k }$

Question 69

Use the vectors u and v to find (3u)× v. $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

A) $( 3 \mathbf { u } ) \times \mathbf { v } = - 141 \mathbf { i } + 165 \mathbf { j } + 144 \mathbf { k }$
B) $\text { (3u) } \times \mathbf { v } = 165 \mathbf { i } - 141 \mathbf { j } + 144 \mathbf { k }$
C) $( 3 \mathbf { u } ) \times \mathbf { v } = 165 \mathbf { i } + 165 \mathbf { j } + 144 \mathbf { k }$
D) $( 3 \mathbf { u } ) \times \mathbf { v } = 165 \mathbf { i } + 144 \mathbf { j } + 144 \mathbf { k }$
E) $\text { (3u) } \times \mathbf { v } = 144 \mathbf { i } + 165 \mathbf { j } - 141 \mathbf { k }$

Question 70

Use the vectors u and v to find u × (2v). $\mathbf { u } = 6 \mathbf { i } - \mathbf { j } + 7 \mathbf { k } \quad \mathbf { v } = 5 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }$

A) $\mathbf { u } \times ( 2 \mathbf { v } ) = - 68 \mathbf { i } + 82 \mathbf { j } + 82 \mathbf { k }$
B) $\mathbf { u } \times ( 2 \mathbf { v } ) = - 68 \mathbf { i } + 82 \mathbf { j } + 70 \mathbf { k }$
C) $\mathbf { u } \times ( 2 \mathbf { v } ) = - 68 \mathbf { i } + 70 \mathbf { j } + 70 \mathbf { k }$
D) $\mathbf { u } \times ( 2 \mathbf { v } ) = 82 \mathbf { i } + 82 \mathbf { j } + 70 \mathbf { k }$
E) $\mathbf { u } \times ( 2 \mathbf { v } ) = 70 \mathbf { i } + 82 \mathbf { j } + 70 \mathbf { k }$

Question 71

Find u × v and show that it is orthogonal to both u and v. $\begin{array} { l } \mathbf { u } = 8 \mathbf { k } \\\mathbf { v } = - \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\end{array}$

A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 4 \mathbf { i } - 8 \mathbf { j } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { j } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 32 \mathbf { j } - 8 \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 8 \mathbf { i } - 8 \mathbf { j } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$

Question 72

Find u × v and show that it is orthogonal to both u and v. $\begin{array} { l } \mathbf { u } = - \mathbf { i } + \mathbf { k } \\\mathbf { v } = \mathbf { j } - 3 \mathbf { k }\end{array}$

A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = \mathbf { i } - 3 \mathbf { j } - \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - \mathbf { i } - 3 \mathbf { j } - \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - \mathbf { i } - 3 \mathbf { j } + \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$
E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0\end{array}$

Question 73

Use the vectors u and v to find u × (-v). $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

A) $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$
B) $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$
C) $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } + 47 \mathbf { j } - 48 \mathbf { k }$
D) $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } + 47 \mathbf { j } + 48 \mathbf { k }$
E) $\mathbf { u } \times ( - \mathbf { v } ) = 48 \mathbf { i } - 55 \mathbf { j } + 47 \mathbf { k }$

Question 74

Use the vectors u and v to find (-2u)× v. $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

A) $( - 2 \mathbf { u } ) \times \mathbf { v } = - 14 \mathbf { i } - 22 \mathbf { j } - 16 \mathbf { k }$
B) $( - 2 \mathbf { u } ) \times \mathbf { v } = - 16 \mathbf { i } - 22 \mathbf { j } + 14 \mathbf { k }$
C) $( - 2 \mathbf { u } ) \times \mathbf { v } = - 16 \mathbf { i } + 14 \mathbf { j } - 22 \mathbf { k }$
D) $( - 2 \mathbf { u } ) \times \mathbf { v } = - 22 \mathbf { i } + 14 \mathbf { j } - 16 \mathbf { k }$
E) $( - 2 \mathbf { u } ) \times \mathbf { v } = 14 \mathbf { i } - 22 \mathbf { j } - 16 \mathbf { k }$

Question 75

Find the distance between the point and the plane.
(-5,-6,-5) $$2 x + 3 y - 9 z = - 1$$

A)18
B) $\frac { 18 } { 94 }$
C) $\frac { 1 } { \sqrt { 94 } }$
D)0
E) $\frac { 18 } { \sqrt { 94 } }$

Question 76

Find the angle,in degrees,between two adjacent sides of the pyramid shown below.Round to the nearest tenth of a degree.[Note: The base of the pyramid is not considered a side.] P(10,0,0),Q(10,10,0),R(0,10,0),S(5,5,2)

A)2.6°
B)47.1°
C)90°
D)45°
E)59.5°

Question 77

Find a set of parametric equations for the line through the point and parallel to the specified line.

Question 78

Use the vectors u and v to find v × (u × u). $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

A)v × (u × u)= -12
B)v × (u × u)= 0
C)v × (u × u)= 12
D)v × (u × u)= -11
E)v × (u × u)= 11

Question 79

Find a set of parametric equations for the line that passes through the given points.
(8,2,3), (-1,3,-6)

Question 80

Use the vectors v to find v×v. $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

A)v × v = -8
B)v × v = -7
C)v × v = 0
D)v × v = 8
E)v × v = 7