Question 1

Find u × v.  $\mathbf { u } = ( - 5 , - 1 , - 1 ) , \quad \mathbf { v } = ( 4,3,0 )$

Accepted Answer

A)  $( 3 , - 4 , - 11 )$ 
B) -23 
C) -17 
D)  $( 3,4 , - 11 )$ 
E)  $( - 20 , - 3,0 )$ 
A)  $( 3 , - 4 , - 11 )$ 
B) -23 
C) -17 
D)  $( 3,4 , - 11 )$ 
E)  $( - 20 , - 3,0 )$

Question 2

Find the distance between the points.  $( 2,1,4 ) , ( 10,2,7 )$

Accepted Answer

A)  $\sqrt { 78 }$ units 
B)  $5 \sqrt { 3 }$ units 
C)  $\sqrt { 74 }$ units 
D)  $\sqrt { 77 }$ units 
E)  $2 \sqrt { 19 }$ units 
A)  $\sqrt { 78 }$ units 
B)  $5 \sqrt { 3 }$ units 
C)  $\sqrt { 74 }$ units 
D)  $\sqrt { 77 }$ units 
E)  $2 \sqrt { 19 }$ units

Question 3

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: (-7,0,0), (-3,2,4)

Accepted Answer

A)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 36$ 
B)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 18$ 
C)  $( x - 5 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 9$ 
D)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 9$ 
E)  $( x + 5 ) ^ { 2 } - ( y - 1 ) ^ { 2 } - ( z - 2 ) ^ { 2 } = 9$ 
A)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 36$ 
B)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 18$ 
C)  $( x - 5 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 9$ 
D)  $( x + 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 9$ 
E)  $( x + 5 ) ^ { 2 } - ( y - 1 ) ^ { 2 } - ( z - 2 ) ^ { 2 } = 9$

Question 4

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.

Accepted Answer

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 
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 5

Find the center of the sphere.  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 4 y + 9 = 0$

Accepted Answer

A) Center: $( 3 , - 2,0 )$ 
B) Center: $( - 3 , - 2,0 )$ 
C) Center: $( - 3,2,0 )$ 
D) Center: $( 0 , - 2,0 )$ 
E) Center: $( 3,0,0 )$ 
A) Center: $( 3 , - 2,0 )$ 
B) Center: $( - 3 , - 2,0 )$ 
C) Center: $( - 3,2,0 )$ 
D) Center: $( 0 , - 2,0 )$ 
E) Center: $( 3,0,0 )$

Question 6

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)

Accepted Answer

A) 2.6° 
B) 47.1° 
C) 90° 
D) 45° 
E) 59.5° 
A) 2.6° 
B) 47.1° 
C) 90° 
D) 45° 
E) 59.5°

Question 7

Find the area of the parallelogram that has the vectors as adjacent sides.  $\begin{array} { l } 
\mathbf { u } = - 3 \mathbf { i } + 6 \mathbf { j } + 9 \mathbf { k } \
\mathbf { v } = 6 \mathbf { i } + 4 \mathbf { j } + 13 \mathbf { k }
\end{array}$

Accepted Answer

A)  $\text { Area } = \sqrt { 8109 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 12717 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 10911 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 10461 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 1905 } \text { square units }$ 
A)  $\text { Area } = \sqrt { 8109 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 12717 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 10911 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 10461 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 1905 } \text { square units }$

Question 8

Use vectors to determine whether the points are collinear. (5,3,-5), (9,1,-1), (-11,11,-21)

Accepted Answer

A) collinear 
B) not collinear 
A) collinear 
B) not collinear

Question 9

Determine whether u and v are parallel,orthogonal,or neither. u = $( 7,0,2 )$ ,v =  $( - 8,0,1 )$

Accepted Answer

A) parallel 
B) neither 
C) orthogonal 
A) parallel 
B) neither 
C) orthogonal

Question 10

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

Accepted Answer

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)  $\begin{array} { l } 
x = 5 + t \
y = 6 + t \
z = 6
\end{array}$ 
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 11

The vector v and its initial point are given.Find the terminal point.  $\mathbf { v } = \left( 4 , \frac { 3 } { 2 } , - \frac { 1 } { 4 } \right)$
Initial point: $\left( 2,1 , - \frac { 3 } { 2 } \right)$

Accepted Answer

A) Terminal point is ( $\frac { 5 } { 2 }$ ,6, $- \frac { 7 } { 4 }$ ). 
B) Terminal point is ( $- \frac { 7 } { 4 }$ , $\frac { 5 } { 2 }$ ,6). 
C) Terminal point is (0, $\frac { 5 } { 2 }$ , $- \frac { 7 } { 4 }$ ). 
D) Terminal point is (6, $\frac { 5 } { 2 }$ , $- \frac { 7 } { 4 }$ ). 
E) Terminal point is (6,6,6). 
A) Terminal point is ( $\frac { 5 } { 2 }$ ,6, $- \frac { 7 } { 4 }$ ). 
B) Terminal point is ( $- \frac { 7 } { 4 }$ , $\frac { 5 } { 2 }$ ,6). 
C) Terminal point is (0, $\frac { 5 } { 2 }$ , $- \frac { 7 } { 4 }$ ). 
D) Terminal point is (6, $\frac { 5 } { 2 }$ , $- \frac { 7 } { 4 }$ ). 
E) Terminal point is (6,6,6).

Question 12

Verify that the points are the vertices of a parallelogram.  $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$

Accepted Answer

A)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 1,4,5 )$ is parallel to $\overrightarrow { A C } = ( 1,4,5 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram. 
B)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 5,4,1 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 1,2,3 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram. 
C)  $\overrightarrow { A B } = ( 1,2,3 )$ is not parallel to $\overrightarrow { C D } = ( 1,2,3 )$
 $\overrightarrow { B D } = ( 5,4,1 )$ is not parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are not parallel and same length. So $A B C D$ form a parallelogram. 
D)  $\overrightarrow { A B } = ( 1,2,3 )$ is perpendicular to $\overrightarrow { C D } = ( 1,2,3 )$ 
$\overrightarrow { B D } = ( 5,4,1 )$ is perpendicular to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are perpendicular and same length. So $A B C D$ form a parallelogram . 
E)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram. 
A)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 1,4,5 )$ is parallel to $\overrightarrow { A C } = ( 1,4,5 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram. 
B)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 5,4,1 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 1,2,3 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram. 
C)  $\overrightarrow { A B } = ( 1,2,3 )$ is not parallel to $\overrightarrow { C D } = ( 1,2,3 )$
 $\overrightarrow { B D } = ( 5,4,1 )$ is not parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are not parallel and same length. So $A B C D$ form a parallelogram. 
D)  $\overrightarrow { A B } = ( 1,2,3 )$ is perpendicular to $\overrightarrow { C D } = ( 1,2,3 )$ 
$\overrightarrow { B D } = ( 5,4,1 )$ is perpendicular to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are perpendicular and same length. So $A B C D$ form a parallelogram . 
E)  $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.

Question 13

Find the distance between the points. $( 5,3 , - 4 ) , ( - 6 , - 3 , - 8 )$

Accepted Answer

A)  $5 \sqrt { 7 }$ units 
B)  $\sqrt { 177 }$ units 
C)  $\sqrt { 174 }$ units 
D)  $4 \sqrt { 11 }$ units 
E)  $\sqrt { 173 }$ units 
A)  $5 \sqrt { 7 }$ units 
B)  $\sqrt { 177 }$ units 
C)  $\sqrt { 174 }$ units 
D)  $4 \sqrt { 11 }$ units 
E)  $\sqrt { 173 }$ units

Question 14

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $x y < - 5$

Accepted Answer

A) Octant II 
B) Octants I,II,III,IV 
C) Octants II,IV,VI,VIII 
D) Octants II,VI,VIII 
E) Octants I,II,III 
A) Octant II 
B) Octants I,II,III,IV 
C) Octants II,IV,VI,VIII 
D) Octants II,VI,VIII 
E) Octants I,II,III

Question 15

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( 5 , - 3,2 ) , \quad \mathbf { v } = ( - 3,3 , - 3 )$

Accepted Answer

A)  $\sqrt { 126 }$ 
B)  $\sqrt { 1 }$ 
C)  $24$ 
D)  $\sqrt { 30 }$ 
E) 0 
A)  $\sqrt { 126 }$ 
B)  $\sqrt { 1 }$ 
C)  $24$ 
D)  $\sqrt { 30 }$ 
E) 0

Question 16

Find the area of the triangle with the given vertices. $A ( 8 , - 3 , - 3 ) , B ( 8 , - 8 , - 9 ) , C ( - 2,6 , - 6 )$

Accepted Answer

A)  $\sqrt { 2715.25 }$ 
B)  $2 \sqrt { 2715.25 }$ 
C)  $\frac { \sqrt { 2715.25 } } { 2 }$ 
D) $4 \sqrt { 2715.25 }$ 
E)  $\frac { \sqrt { 2715.25 } } { 4 }$ 
A)  $\sqrt { 2715.25 }$ 
B)  $2 \sqrt { 2715.25 }$ 
C)  $\frac { \sqrt { 2715.25 } } { 2 }$ 
D) $4 \sqrt { 2715.25 }$ 
E)  $\frac { \sqrt { 2715.25 } } { 4 }$

Question 17

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

Accepted Answer

A)  $\mathbf { u } ( \mathbf { u } \times \mathbf { v } ) = 10$ 
B)  $\mathbf { u } ( \mathbf { u } \times \mathbf { v } ) = - 10$ 
C)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = 9$ 
D)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = - 9$ 
E)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = 0$ 
A)  $\mathbf { u } ( \mathbf { u } \times \mathbf { v } ) = 10$ 
B)  $\mathbf { u } ( \mathbf { u } \times \mathbf { v } ) = - 10$ 
C)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = 9$ 
D)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = - 9$ 
E)  $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } ) = 0$

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
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 20

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

Accepted Answer

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

Find u × v. $\mathbf { u } = ( - 5 , - 1 , - 1 ) , \quad \mathbf { v } = ( 4,3,0 )$

Find the distance between the points. $( 2,1,4 ) , ( 10,2,7 )$

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: (-7,0,0), (-3,2,4)

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.

Find the center of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 4 y + 9 = 0$

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)

Find the area of the parallelogram that has the vectors as adjacent sides. =-3+6+9 =6+4+13

Use vectors to determine whether the points are collinear. (5,3,-5), (9,1,-1), (-11,11,-21)

Determine whether u and v are parallel,orthogonal,or neither. u = $( 7,0,2 )$ ,v = $( - 8,0,1 )$

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

The vector v and its initial point are given.Find the terminal point. $\mathbf { v } = \left( 4 , \frac { 3 } { 2 } , - \frac { 1 } { 4 } \right)$ Initial point: $\left( 2,1 , - \frac { 3 } { 2 } \right)$

Verify that the points are the vertices of a parallelogram. $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$

Find the distance between the points. $( 5,3 , - 4 ) , ( - 6 , - 3 , - 8 )$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $x y < - 5$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( 5 , - 3,2 ) , \quad \mathbf { v } = ( - 3,3 , - 3 )$

Find the area of the triangle with the given vertices. $A ( 8 , - 3 , - 3 ) , B ( 8 , - 8 , - 9 ) , C ( - 2,6 , - 6 )$

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

Find the angle between the two planes. 3x-4y+5z=6 x+y-z=2

Find u × v and show that it is orthogonal to both u and v. =-+ =-3

Find u × v and show that it is orthogonal to both u and v. =-3 =-+

Functions and Their Graphs

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Exam 11: Analytic Geometry In Three Dimensions

Find u × v. u=(−5,−1,−1),v=(4,3,0)\mathbf { u } = ( - 5 , - 1 , - 1 ) , \quad \mathbf { v } = ( 4,3,0 )u=(−5,−1,−1),v=(4,3,0)

Find the distance between the points. (2,1,4),(10,2,7)( 2,1,4 ) , ( 10,2,7 )(2,1,4),(10,2,7)

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: (-7,0,0), (-3,2,4)

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.

Find the center of the sphere. x2+y2+z2−6x+4y+9=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 4 y + 9 = 0x2+y2+z2−6x+4y+9=0

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)

Find the area of the parallelogram that has the vectors as adjacent sides. =-3+6+9 =6+4+13

Use vectors to determine whether the points are collinear. (5,3,-5), (9,1,-1), (-11,11,-21)

Determine whether u and v are parallel,orthogonal,or neither. u = (7,0,2)( 7,0,2 )(7,0,2) ,v = (−8,0,1)( - 8,0,1 )(−8,0,1)

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

The vector v and its initial point are given.Find the terminal point. v=(4,32,−14)\mathbf { v } = \left( 4 , \frac { 3 } { 2 } , - \frac { 1 } { 4 } \right)v=(4,23​,−41​) Initial point: (2,1,−32)\left( 2,1 , - \frac { 3 } { 2 } \right)(2,1,−23​)

Verify that the points are the vertices of a parallelogram. A(2,2,2),B(3,4,5),C(7,6,3),D(8,8,6)A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )A(2,2,2),B(3,4,5),C(7,6,3),D(8,8,6)

Find the distance between the points. (5,3,−4),(−6,−3,−8)( 5,3 , - 4 ) , ( - 6 , - 3 , - 8 )(5,3,−4),(−6,−3,−8)

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. xy<−5x y < - 5xy<−5

Find the area of the parallelogram that has the vectors as adjacent sides. u=(5,−3,2),v=(−3,3,−3)\mathbf { u } = ( 5 , - 3,2 ) , \quad \mathbf { v } = ( - 3,3 , - 3 )u=(5,−3,2),v=(−3,3,−3)

Find the area of the triangle with the given vertices. A(8,−3,−3),B(8,−8,−9),C(−2,6,−6)A ( 8 , - 3 , - 3 ) , B ( 8 , - 8 , - 9 ) , C ( - 2,6 , - 6 )A(8,−3,−3),B(8,−8,−9),C(−2,6,−6)

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

Find the angle between the two planes. 3x-4y+5z=6 x+y-z=2

Find u × v and show that it is orthogonal to both u and v. =-+ =-3

Find u × v and show that it is orthogonal to both u and v. =-3 =-+

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Find u × v. $\mathbf { u } = ( - 5 , - 1 , - 1 ) , \quad \mathbf { v } = ( 4,3,0 )$

Find the distance between the points. $( 2,1,4 ) , ( 10,2,7 )$

Find the center of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x + 4 y + 9 = 0$

Determine whether u and v are parallel,orthogonal,or neither. u = $( 7,0,2 )$ ,v = $( - 8,0,1 )$

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

The vector v and its initial point are given.Find the terminal point. $\mathbf { v } = \left( 4 , \frac { 3 } { 2 } , - \frac { 1 } { 4 } \right)$ Initial point: $\left( 2,1 , - \frac { 3 } { 2 } \right)$

Verify that the points are the vertices of a parallelogram. $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$

Find the distance between the points. $( 5,3 , - 4 ) , ( - 6 , - 3 , - 8 )$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $x y < - 5$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( 5 , - 3,2 ) , \quad \mathbf { v } = ( - 3,3 , - 3 )$

Find the area of the triangle with the given vertices. $A ( 8 , - 3 , - 3 ) , B ( 8 , - 8 , - 9 ) , C ( - 2,6 , - 6 )$

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