Question 1

Find the dot product v · w.
-v = -i - j, w = i + j

Accepted Answer

A)  -2 
B)  1 
C)  -1 
D)  0 
A)  -2 
B)  1 
C)  -1 
D)  0

Question 2

Plot the point given in polar coordinates.
-$$\left( 4 , \frac { - 5 \pi } { 4 } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 3

Plot the point given in polar coordinates.
-$$\left( 4,30 ^ { \circ } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 4

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
-$$x ^ { 2 } + 4 y ^ { 2 } = 4$$

Accepted Answer

A)  $4 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 4 \mathrm { r }$ 
B)  $\cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta = 4 \mathrm { r }$ 
C)  $r ^ { 2 } \left( \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta \right) = 4$ 
D)  $\mathrm { r } ^ { 2 } \left( 4 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) = 4$ 
A)  $4 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 4 \mathrm { r }$ 
B)  $\cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta = 4 \mathrm { r }$ 
C)  $r ^ { 2 } \left( \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta \right) = 4$ 
D)  $\mathrm { r } ^ { 2 } \left( 4 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) = 4$

Question 5

Find the area of the parallelogram with vertices $$\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \text { and } \mathrm { P } _ { 4 }$$ .
-$$P _ { 1 } = ( 7 , - 5,5 ) , \quad P _ { 2 } = ( 4,7 , - 5 ) , \quad P _ { 3 } = ( 6,4 , - 2 ) , \quad P _ { 4 } = ( 3,16 , - 12 )$$

Accepted Answer

A)  $\sqrt { 32758 }$ 
B)  $\frac { \sqrt { 382 } } { 2 }$ 
C)  $\sqrt { 382 }$ 
D)  $\frac { \sqrt { 32758 } } { 2 }$ 
A)  $\sqrt { 32758 }$ 
B)  $\frac { \sqrt { 382 } } { 2 }$ 
C)  $\sqrt { 382 }$ 
D)  $\frac { \sqrt { 32758 } } { 2 }$

Question 6

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).
-$$r = 5$$

Accepted Answer

A)  $x + y = 5$ 
B)  $x ^ { 2 } - y ^ { 2 } = 25$ 
C)  $x ^ { 2 } + y ^ { 2 } = 25$ 
D)  $x + y = 25$ 
A)  $x + y = 5$ 
B)  $x ^ { 2 } - y ^ { 2 } = 25$ 
C)  $x ^ { 2 } + y ^ { 2 } = 25$ 
D)  $x + y = 25$

Question 7

Solve the problem.
-At a state fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in the figure. One of the pickups pulls with a force of 1,400 pounds and the other pulls with a force of 3,400 pounds
With an angle of 45° between them. With how much force must the monster truck pull in order to remain
Unmoved? HINT: Find the resultant force of the two trucks. Round your answer to the nearest tenth.

Accepted Answer

A)  The truck must pull with a force of 2,605.4 lb. 
B)  The truck must pull with a force of 4,500.2 lb. 
C)  The truck must pull with a force of 4,276.9 lb. 
D)  The truck must pull with a force of 2,197.3 lb. 
A)  The truck must pull with a force of 2,605.4 lb. 
B)  The truck must pull with a force of 4,500.2 lb. 
C)  The truck must pull with a force of 4,276.9 lb. 
D)  The truck must pull with a force of 2,197.3 lb.

Question 8

Find the unit vector having the same direction as v.
-$$v = - 12 i + 5 j$$

Accepted Answer

A)  $\mathbf { u } = - \frac { 12 } { 13 } \mathbf { i } + \frac { 5 } { 13 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { 13 } { 12 } \mathbf { i } + \frac { 13 } { 5 } \mathbf { j }$ 
C)  $u = - \frac { 5 } { 13 } \mathrm { i } + \frac { 12 } { 13 } \mathrm { j }$ 
D)  $\mathbf { u } = - 156 \mathbf { i } + 65 \mathrm { j }$ 
A)  $\mathbf { u } = - \frac { 12 } { 13 } \mathbf { i } + \frac { 5 } { 13 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { 13 } { 12 } \mathbf { i } + \frac { 13 } { 5 } \mathbf { j }$ 
C)  $u = - \frac { 5 } { 13 } \mathrm { i } + \frac { 12 } { 13 } \mathrm { j }$ 
D)  $\mathbf { u } = - 156 \mathbf { i } + 65 \mathrm { j }$

Question 9

Find the unit vector having the same direction as v.
-$$v = - 3 i + j$$

Accepted Answer

A)  $\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathrm { i } - \frac { \sqrt { 10 } } { 10 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { \sqrt { 10 } } { 3 } \mathbf { i } + \sqrt { 10 } \mathbf { j }$ 
C)  $\mathbf { u } = - 3 \sqrt { 10 } \mathbf { i } + \sqrt { 10 } \mathbf { j }$ 
D)  $\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathbf { i } + \frac { \sqrt { 10 } } { 10 } \mathbf { j }$ 
A)  $\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathrm { i } - \frac { \sqrt { 10 } } { 10 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { \sqrt { 10 } } { 3 } \mathbf { i } + \sqrt { 10 } \mathbf { j }$ 
C)  $\mathbf { u } = - 3 \sqrt { 10 } \mathbf { i } + \sqrt { 10 } \mathbf { j }$ 
D)  $\mathbf { u } = - \frac { 3 \sqrt { 10 } } { 10 } \mathbf { i } + \frac { \sqrt { 10 } } { 10 } \mathbf { j }$

Question 10

Solve the problem. Round your answer to the nearest tenth.
-A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Accepted Answer

A)  1617.4 ft-lb 
B)  1267.9 ft-lb 
C)  2110.8 ft-lb 
D)  2718.9 ft-lb 
A)  1617.4 ft-lb 
B)  1267.9 ft-lb 
C)  2110.8 ft-lb 
D)  2718.9 ft-lb

Question 11

Solve the problem. Round your answer to the nearest tenth.
-Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75 feet from (0, 0) to (-75, 0).

Accepted Answer

A)  8944.9 ft-lb 
B)  13,416.1 ft-lb 
C)  6708.2 ft-lb 
D)  15,000.0 ft-lb 
A)  8944.9 ft-lb 
B)  13,416.1 ft-lb 
C)  6708.2 ft-lb 
D)  15,000.0 ft-lb

Question 12

Use the given vectors to find the indicated expression.
-v = -3i + 2j + k, w = -4i + 3j - k Find (-3v) × w.

Accepted Answer

A)  3i + 5j + 35k 
B)  15i + 21j + 3k 
C)  -15i + 27j - 3k 
D)  7i + 9j - 33k 
A)  3i + 5j + 35k 
B)  15i + 21j + 3k 
C)  -15i + 27j - 3k 
D)  7i + 9j - 33k

Question 13

Find the angle between v and w. Round your answer to one decimal place, if necessary.
-v = 4i + 7j, w = 3i + 8j

Accepted Answer

A)  354.6° 
B)  9.2° 
C)  4.6° 
D)  19.2° 
A)  354.6° 
B)  9.2° 
C)  4.6° 
D)  19.2°

Question 14

Solve the problem.
-Which of the following vectors is parallel to v = 9i + 5j? A) $\mathbf { w } = - \frac { 15 } { 2 } \mathrm { i } + \frac { 25 } { 6 } \mathrm { j }$

Accepted Answer

B)  $w = 45 i - 25 j$ 
C)  $\mathbf { w } = \frac { 15 } { 2 } \mathrm { i } - \frac { 25 } { 6 } \mathrm { j }$ 
D)  $\mathbf { w } = \frac { 3 } { 2 } i + \frac { 5 } { 6 } \mathbf { j }$ 
B)  $w = 45 i - 25 j$ 
C)  $\mathbf { w } = \frac { 15 } { 2 } \mathrm { i } - \frac { 25 } { 6 } \mathrm { j }$ 
D)  $\mathbf { w } = \frac { 3 } { 2 } i + \frac { 5 } { 6 } \mathbf { j }$

Question 15

Perform the indicated operation.
-$$\begin{array} { l } 
\mathbf { v } = - 5 \mathbf { i } - 4 \mathbf { j } - 3 \mathbf { k } \text { and } \mathbf { w } = 3 \mathbf { i } - 2 \mathbf { j } + 6 \mathbf { k } \
\text { Find } \| \mathbf { v } - \mathbf { w } \| \
\begin{array} { l l l l } 
\text { A) } \sqrt { 161 } & \text { B) } \sqrt { 110 } & \text { C) } \sqrt { 149 } & \text { D) } 5 \sqrt { 2 } - 7
\end{array}
\end{array}$$

Accepted Answer

Question 16

State whether the vectors are parallel, orthogonal, or neither.
-$$\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathrm { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathrm { j }$$

Accepted Answer

A)  Parallel 
B)  Orthogonal 
C)  Neither 
A)  Parallel 
B)  Orthogonal 
C)  Neither

Question 17

Find the distance from $$\mathrm { P } _ { 1 } \text { to } \mathrm { P } _ { 2 }$$
-$$\mathrm { P } _ { 1 } = ( 0 , - 4,4 ) \text { and } \mathrm { P } _ { 2 } = ( - 1,1 , - 3 )$$

Accepted Answer

A)  $\sqrt { 74 }$ 
B)  $5 \sqrt { 3 }$ 
C)  $2 \sqrt { 10 }$ 
D)  $\sqrt { 11 }$ 
A)  $\sqrt { 74 }$ 
B)  $5 \sqrt { 3 }$ 
C)  $2 \sqrt { 10 }$ 
D)  $\sqrt { 11 }$

Question 18

Find the indicated cross product.
-v = -6i - 3j + k, w = -3i - 5j - k Find v × w.

Accepted Answer

A)  8i - 9j + 21k 
B)  21i - 8j + 9k 
C)  -8i + 9j + 3k 
D)  -2i - 3j + 39k 
A)  8i - 9j + 21k 
B)  21i - 8j + 9k 
C)  -8i + 9j + 3k 
D)  -2i - 3j + 39k

Question 19

Find the value of the determinant.
-$$\left| \begin{array} { l l } 
8 & 5 \
7 & 7
\end{array} \right|$$

Accepted Answer

A)  91 
B)  -21 
C)  21 
D)  -9 
A)  91 
B)  -21 
C)  21 
D)  -9

Question 20

Find the position vector for the vector having initial point P and terminal point Q.
-P = (4, 2, 2) and Q = (-4, -1, -2)

Accepted Answer

A)  v = -6i - 3j - 6k 
B)  v = -8i - 3j - 4k 
C)  v = -6i - 3j - 2k 
D)  v = 8i + 3j + 4k 
A)  v = -6i - 3j - 6k 
B)  v = -8i - 3j - 4k 
C)  v = -6i - 3j - 2k 
D)  v = 8i + 3j + 4k

Find the dot product v · w. -v = -i - j, w = i + j

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $x ^ { 2 } + 4 y ^ { 2 } = 4$

Find the area of the parallelogram with vertices $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \text { and } \mathrm { P } _ { 4 }$ . - $P _ { 1 } = ( 7 , - 5,5 ) , \quad P _ { 2 } = ( 4,7 , - 5 ) , \quad P _ { 3 } = ( 6,4 , - 2 ) , \quad P _ { 4 } = ( 3,16 , - 12 )$

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - $r = 5$

Find the unit vector having the same direction as v. - $v = - 12 i + 5 j$

Find the unit vector having the same direction as v. - $v = - 3 i + j$

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Solve the problem. Round your answer to the nearest tenth. -Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75 feet from (0, 0) to (-75, 0).

Use the given vectors to find the indicated expression. -v = -3i + 2j + k, w = -4i + 3j - k Find (-3v) × w.

Find the angle between v and w. Round your answer to one decimal place, if necessary. -v = 4i + 7j, w = 3i + 8j

Perform the indicated operation. - =-5-4-3 and =3-2+6 Find \|-\| A) B) C) D) 5-7

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathrm { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathrm { j }$

Find the distance from $\mathrm { P } _ { 1 } \text { to } \mathrm { P } _ { 2 }$ - $\mathrm { P } _ { 1 } = ( 0 , - 4,4 ) \text { and } \mathrm { P } _ { 2 } = ( - 1,1 , - 3 )$

Find the indicated cross product. -v = -6i - 3j + k, w = -3i - 5j - k Find v × w.

Find the value of the determinant. - $\left| \begin{array} { l l } 8 & 5 \\7 & 7\end{array} \right|$

Find the position vector for the vector having initial point P and terminal point Q. -P = (4, 2, 2) and Q = (-4, -1, -2)

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Exam 8: Polar Coordinates; Vectors