Question 1

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

Accepted Answer

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

Question 2

Find the standard form of the equation of the sphere with the given characteristics. 
Center: $( 5,0 , - 6 )$ ;radius: 6

Accepted Answer

A)  $( x - 5 ) ^ { 2 } - y ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
B)  $( x - 5 ) ^ { 2 } - y ^ { 2 } - ( z + 6 ) ^ { 2 } = 36$ 
C)  $( x - 5 ) ^ { 2 } + y ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
D)  $( x - 5 ) ^ { 2 } + y ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
E)  $( x - 5 ) ^ { 2 } - y ^ { 2 } - ( z - 6 ) ^ { 2 } = 36$ 
A)  $( x - 5 ) ^ { 2 } - y ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
B)  $( x - 5 ) ^ { 2 } - y ^ { 2 } - ( z + 6 ) ^ { 2 } = 36$ 
C)  $( x - 5 ) ^ { 2 } + y ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
D)  $( x - 5 ) ^ { 2 } + y ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
E)  $( x - 5 ) ^ { 2 } - y ^ { 2 } - ( z - 6 ) ^ { 2 } = 36$

Question 3

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

Accepted Answer

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 
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 4

Find a set of parametric equations for the line through the point and parallel to the specified vector. $( - 5,6 , - 8 ) \text {, parallel to } ( 6,5 , - 9 )$

Accepted Answer

Answers may vary.One

Question 5

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: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

Accepted Answer

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:  $\frac { x - 4 } { 2 } = \frac { y + 4 } { - 4 } = z - 5$ 
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 6

The vector v and its initial point are given.Find the terminal point.  $\mathbf { v } = \left( \frac { 9 } { 2 } , - \frac { 1 } { 2 } , 8 \right)$
Initial point: $\left( 7,6 , - \frac { 1 } { 2 } \right)$

Accepted Answer

A) Terminal point is $\left( \frac { 23 } { 2 } , \frac { 11 } { 2 } , \frac { 15 } { 2 } \right)$ . 
B) Terminal point is $\left( \frac { 23 } { 2 } , \frac { 15 } { 2 } , \frac { 11 } { 2 } \right)$ . 
C) Terminal point is $\left( \frac { 11 } { 2 } , \frac { 23 } { 2 } , \frac { 15 } { 2 } \right)$ . 
D) Terminal point is $\left( \frac { 11 } { 2 } , \frac { 15 } { 2 } , \frac { 23 } { 2 } \right)$ . 
E) Terminal point is $\left( \frac { 15 } { 2 } , \frac { 11 } { 2 } , \frac { 23 } { 2 } \right)$ . 
A) Terminal point is $\left( \frac { 23 } { 2 } , \frac { 11 } { 2 } , \frac { 15 } { 2 } \right)$ . 
B) Terminal point is $\left( \frac { 23 } { 2 } , \frac { 15 } { 2 } , \frac { 11 } { 2 } \right)$ . 
C) Terminal point is $\left( \frac { 11 } { 2 } , \frac { 23 } { 2 } , \frac { 15 } { 2 } \right)$ . 
D) Terminal point is $\left( \frac { 11 } { 2 } , \frac { 15 } { 2 } , \frac { 23 } { 2 } \right)$ . 
E) Terminal point is $\left( \frac { 15 } { 2 } , \frac { 11 } { 2 } , \frac { 23 } { 2 } \right)$ .

Question 7

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

Accepted Answer

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)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 41 \mathbf { i } + 73 \mathbf { j } - 41 \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
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 8

Find the coordinates of the point. 
The point is located five units behind the yz-plane,six units to the right of the xz-plane,and seven units above the xy-plane.

Accepted Answer

A)  $x = - 5 , y = 6 , z = 7$ 
B)  $x = - 5 , y = 6 , z = - 7$ 
C)  $x = 6 , y = - 5 , z = 7$ 
D)  $x = 7 , y = - 5 , z = 6$ 
E)  $x = 7 , y = 6 , z = - 5$ 
A)  $x = - 5 , y = 6 , z = 7$ 
B)  $x = - 5 , y = 6 , z = - 7$ 
C)  $x = 6 , y = - 5 , z = 7$ 
D)  $x = 7 , y = - 5 , z = 6$ 
E)  $x = 7 , y = 6 , z = - 5$

Question 9

Find the volume of the parallelepiped with the given vertices.  $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 5,5,0 ) , \mathrm { C } ( 5,0,6 ) , \mathrm { D } ( 0,5,5 ) , \mathrm { E } ( 6,5,6 ) , \mathrm { F } ( 5,5,7 ) , \mathrm { G } ( 5,6,5 ) , \mathrm { H } ( 6,6,7 )$

Accepted Answer

A)  $276 \text { cubic units }$ 
B)  $275 \text { cubic units }$ 
C)  $277 \text { cubic units }$ 
D)  $279 \text { cubic units }$ 
E)  $278 \text { cubic units }$ 
A)  $276 \text { cubic units }$ 
B)  $275 \text { cubic units }$ 
C)  $277 \text { cubic units }$ 
D)  $279 \text { cubic units }$ 
E)  $278 \text { cubic units }$

Question 10

Find the magnitude of v.  $\mathbf { v } = ( - 1,0,3 )$

Accepted Answer

A)  $\| \mathbf { v } \|$ = $\sqrt { 2 }$ 
B)  $\| \mathbf { v } \|$ = $\sqrt { 4.72 }$ 
C)  $\| \mathbf { v } \|$ = $\sqrt { 10 }$ 
D)  $\| \mathbf { v } \|$ = $\sqrt { 2.72 }$ 
E)  $\| \mathbf { v } \|$ = $2 \sqrt { 3 }$ 
A)  $\| \mathbf { v } \|$ = $\sqrt { 2 }$ 
B)  $\| \mathbf { v } \|$ = $\sqrt { 4.72 }$ 
C)  $\| \mathbf { v } \|$ = $\sqrt { 10 }$ 
D)  $\| \mathbf { v } \|$ = $\sqrt { 2.72 }$ 
E)  $\| \mathbf { v } \|$ = $2 \sqrt { 3 }$

Question 11

Find the distance between the points. $( 0 , - 4,0 ) , ( 7,0 , - 7 )$

Accepted Answer

A)  $\sqrt { 118 }$ units 
B)  $3 \sqrt { 13 }$ units 
C)  $\sqrt { 114 }$ units 
D)  $2 \sqrt { 29 }$ units 
E)  $\sqrt { 115 }$ units 
A)  $\sqrt { 118 }$ units 
B)  $3 \sqrt { 13 }$ units 
C)  $\sqrt { 114 }$ units 
D)  $2 \sqrt { 29 }$ units 
E)  $\sqrt { 115 }$ units

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: $( 3 , - 6,2 )$ Parallel to: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

Accepted Answer

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

Question 13

Find a unit vector orthogonal to u and v. $\begin{array} { l } 
\mathbf { u } = \mathbf { i } + \mathbf { j } \
\mathbf { v } = \mathbf { j } + \mathbf { k }
\end{array}$

Accepted Answer

A)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( - \mathbf { i } - \mathbf { j } + \mathbf { k } )$ 
B)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( - \mathrm { i } + \mathbf { j } + \mathbf { k } )$ 
C)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$ 
D)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } - \mathbf { j } + \mathbf { k } )$ 
E)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
A)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( - \mathbf { i } - \mathbf { j } + \mathbf { k } )$ 
B)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( - \mathrm { i } + \mathbf { j } + \mathbf { k } )$ 
C)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$ 
D)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } - \mathbf { j } + \mathbf { k } )$ 
E)  $\text { Unit vector } = \frac { \sqrt { 3 } } { 3 } ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$

Question 14

Assume that Earth is a sphere with a radius of 4500 miles.The center of Earth is placed at the origin of a three-dimensional coordinate system.What is the equation of the sphere?

Accepted Answer

A)  $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
B)  $2 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
D)  $x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
E)  $x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
A)  $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
B)  $2 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
D)  $x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = ( 4500 ) ^ { 2 }$ 
E)  $x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = ( 4500 ) ^ { 2 }$

Question 15

Find a unit vector in the direction of u. u = $( 5,2,10 )$

Accepted Answer

A)  $\frac { 1 } { \sqrt { 17 } } ( 5,10,2 )$ 
B)  $\frac { 1 } { \sqrt { 17 } } ( 5,2,10 )$ 
C)  $\frac { 1 } { \sqrt { 129 } } ( 2,5,10 )$ 
D)  $( 5,2,10 )$ 
E)  $\frac { 1 } { \sqrt { 129 } } ( 5,2,10 )$ 
A)  $\frac { 1 } { \sqrt { 17 } } ( 5,10,2 )$ 
B)  $\frac { 1 } { \sqrt { 17 } } ( 5,2,10 )$ 
C)  $\frac { 1 } { \sqrt { 129 } } ( 2,5,10 )$ 
D)  $( 5,2,10 )$ 
E)  $\frac { 1 } { \sqrt { 129 } } ( 5,2,10 )$

Question 16

Find the coordinates of the point located 2 units in front of the yz-plane,6 units to the left of the xz-plane,and 7 units below the xy-plane.

Accepted Answer

A)  $( 2,2 , - 7 )$ 
B)  $( 2 , - 6 , - 7 )$ 
C)  $( 2 , - 6,6 )$ 
D)  $( 2 , - 11,7 )$ 
E)  $( 2 , - 6 , - 4 )$ 
A)  $( 2,2 , - 7 )$ 
B)  $( 2 , - 6 , - 7 )$ 
C)  $( 2 , - 6,6 )$ 
D)  $( 2 , - 11,7 )$ 
E)  $( 2 , - 6 , - 4 )$

Question 17

Find the dot product of u and v. u = $( - 3 , - 8,5 )$ ,v = $( 2 , - 6 , - 7 )$

Accepted Answer

A)  $( - 14 , - 2 , - 1 )$ 
B) 7 
C) 77 
D) -17 
E)  $( 48 , - 35 , - 6 )$ 
A)  $( - 14 , - 2 , - 1 )$ 
B) 7 
C) 77 
D) -17 
E)  $( 48 , - 35 , - 6 )$

Question 18

Find the coordinates of the point located 2 units in front of the yz-plane,8 units to the left of the xz-plane,and 3 units below the xy-plane.

Accepted Answer

A)  $( 2 , - 8 , - 3 )$ 
B)  $( 2,8 , - 3 )$ 
C)  $( 2 , - 8 , - 2 )$ 
D)  $( - 2 , - 8 , - 2 )$ 
E)  $( 2 , - 2 , - 3 )$ 
A)  $( 2 , - 8 , - 3 )$ 
B)  $( 2,8 , - 3 )$ 
C)  $( 2 , - 8 , - 2 )$ 
D)  $( - 2 , - 8 , - 2 )$ 
E)  $( 2 , - 2 , - 3 )$

Question 19

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

Accepted Answer

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)  $\begin{array} { l } 
x = - 7 + 7 t \
y = 4 + 4 t \
z = 7 + t
\end{array}$ 
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 20

Find the area of the parallelogram that has the vectors as adjacent sides.  $\begin{array} { l } 
\mathbf { u } = ( 5,5 , - 7 ) \
\mathbf { v } = ( 0,5,7 )
\end{array}$

Accepted Answer

A)  $\text { Area } = \sqrt { 130.00 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 5500.00 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 6750.00 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 5560.00 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 6100.00 } \text { square units }$ 
A)  $\text { Area } = \sqrt { 130.00 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 5500.00 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 6750.00 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 5560.00 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 6100.00 } \text { square units }$

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

Find the standard form of the equation of the sphere with the given characteristics. Center: $( 5,0 , - 6 )$ ;radius: 6

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

Find a set of parametric equations for the line through the point and parallel to the specified vector. $( - 5,6 , - 8 ) \text {, parallel to } ( 6,5 , - 9 )$

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: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

The vector v and its initial point are given.Find the terminal point. $\mathbf { v } = \left( \frac { 9 } { 2 } , - \frac { 1 } { 2 } , 8 \right)$ Initial point: $\left( 7,6 , - \frac { 1 } { 2 } \right)$

Find u × v and show that it is orthogonal to both u and v. =12+6+ =+5-6

Find the coordinates of the point. The point is located five units behind the yz-plane,six units to the right of the xz-plane,and seven units above the xy-plane.

Find the volume of the parallelepiped with the given vertices. $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 5,5,0 ) , \mathrm { C } ( 5,0,6 ) , \mathrm { D } ( 0,5,5 ) , \mathrm { E } ( 6,5,6 ) , \mathrm { F } ( 5,5,7 ) , \mathrm { G } ( 5,6,5 ) , \mathrm { H } ( 6,6,7 )$

Find the magnitude of v. $\mathbf { v } = ( - 1,0,3 )$

Find the distance between the points. $( 0 , - 4,0 ) , ( 7,0 , - 7 )$

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: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

Find a unit vector orthogonal to u and v. =+ =+

Assume that Earth is a sphere with a radius of 4500 miles.The center of Earth is placed at the origin of a three-dimensional coordinate system.What is the equation of the sphere?

Find a unit vector in the direction of u. u = $( 5,2,10 )$

Find the coordinates of the point located 2 units in front of the yz-plane,6 units to the left of the xz-plane,and 7 units below the xy-plane.

Find the dot product of u and v. u = $( - 3 , - 8,5 )$ ,v = $( 2 , - 6 , - 7 )$

Find the coordinates of the point located 2 units in front of the yz-plane,8 units to the left of the xz-plane,and 3 units below the xy-plane.

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

Find the area of the parallelogram that has the vectors as adjacent sides. =(5,5,-7) =(0,5,7)

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Exam 11: Analytic Geometry In Three Dimensions

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

Find the standard form of the equation of the sphere with the given characteristics. Center: (5,0,−6)( 5,0 , - 6 )(5,0,−6) ;radius: 6

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

Find a set of parametric equations for the line through the point and parallel to the specified vector. (−5,6,−8), parallel to (6,5,−9)( - 5,6 , - 8 ) \text {, parallel to } ( 6,5 , - 9 )(−5,6,−8), parallel to (6,5,−9)

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

The vector v and its initial point are given.Find the terminal point. v=(92,−12,8)\mathbf { v } = \left( \frac { 9 } { 2 } , - \frac { 1 } { 2 } , 8 \right)v=(29​,−21​,8) Initial point: (7,6,−12)\left( 7,6 , - \frac { 1 } { 2 } \right)(7,6,−21​)

Find u × v and show that it is orthogonal to both u and v. =12+6+ =+5-6

Find the coordinates of the point. The point is located five units behind the yz-plane,six units to the right of the xz-plane,and seven units above the xy-plane.

Find the magnitude of v. v=(−1,0,3)\mathbf { v } = ( - 1,0,3 )v=(−1,0,3)

Find the distance between the points. (0,−4,0),(7,0,−7)( 0 , - 4,0 ) , ( 7,0 , - 7 )(0,−4,0),(7,0,−7)

Find a unit vector orthogonal to u and v. =+ =+

Assume that Earth is a sphere with a radius of 4500 miles.The center of Earth is placed at the origin of a three-dimensional coordinate system.What is the equation of the sphere?

Find a unit vector in the direction of u. u = (5,2,10)( 5,2,10 )(5,2,10)

Find the coordinates of the point located 2 units in front of the yz-plane,6 units to the left of the xz-plane,and 7 units below the xy-plane.

Find the dot product of u and v. u = (−3,−8,5)( - 3 , - 8,5 )(−3,−8,5) ,v = (2,−6,−7)( 2 , - 6 , - 7 )(2,−6,−7)

Find the coordinates of the point located 2 units in front of the yz-plane,8 units to the left of the xz-plane,and 3 units below the xy-plane.

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

Find the area of the parallelogram that has the vectors as adjacent sides. =(5,5,-7) =(0,5,7)

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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

Find the standard form of the equation of the sphere with the given characteristics. Center: $( 5,0 , - 6 )$ ;radius: 6

Find a set of parametric equations for the line through the point and parallel to the specified vector. $( - 5,6 , - 8 ) \text {, parallel to } ( 6,5 , - 9 )$

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: $x = 5 + 2 t , y = 7 - 4 t , z = - 2 + t$

The vector v and its initial point are given.Find the terminal point. $\mathbf { v } = \left( \frac { 9 } { 2 } , - \frac { 1 } { 2 } , 8 \right)$ Initial point: $\left( 7,6 , - \frac { 1 } { 2 } \right)$

Find the magnitude of v. $\mathbf { v } = ( - 1,0,3 )$

Find the distance between the points. $( 0 , - 4,0 ) , ( 7,0 , - 7 )$

Find a unit vector in the direction of u. u = $( 5,2,10 )$

Find the dot product of u and v. u = $( - 3 , - 8,5 )$ ,v = $( 2 , - 6 , - 7 )$

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