Question 1

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = 10 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) \
w = 5 \left( \cos 10 ^ { \circ } + i \sin 10 ^ { \circ } \right)
\end{array}$$
Find $\frac { Z } { w }$.

Accepted Answer

A)  $2 \left( \cos 3 ^ { \circ } + i \sin 3 ^ { \circ } \right)$ 
B)  $5 \left( \cos 3 ^ { \circ } + i \sin 3 ^ { \circ } \right)$ 
C)  $5 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right)$ 
D)  $2 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right)$ 
A)  $2 \left( \cos 3 ^ { \circ } + i \sin 3 ^ { \circ } \right)$ 
B)  $5 \left( \cos 3 ^ { \circ } + i \sin 3 ^ { \circ } \right)$ 
C)  $5 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right)$ 
D)  $2 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right)$

Question 2

Plot the point given in polar coordinates.
-$$\left( - 2 , \frac { 9 \pi } { 4 } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 3

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).
-$$r = \frac { 5 } { 1 + \cos \theta }$$

Accepted Answer

A)  $x ^ { 2 } = 25 - 10 y$ 
B)  $y ^ { 2 } = 25 - 10 x$ 
C)  $x ^ { 2 } = 10 y - 25$ 
D)  $\mathrm { y } ^ { 2 } = 10 \mathrm { x } - 25$ 
A)  $x ^ { 2 } = 25 - 10 y$ 
B)  $y ^ { 2 } = 25 - 10 x$ 
C)  $x ^ { 2 } = 10 y - 25$ 
D)  $\mathrm { y } ^ { 2 } = 10 \mathrm { x } - 25$

Question 4

Write the expression in the standard form a + bi.
-$\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }$

Accepted Answer

A)  $- \frac { 9 } { 2 } - \frac { 9 \sqrt { 3 } } { 2 } i$ 
B)  $- \frac { 9 } { 2 } + \frac { 9 \sqrt { 3 } } { 2 } i$ 
C)  $- \frac { 9 \sqrt { 3 } } { 2 } + \frac { 9 } { 2 } \mathrm { i }$ 
D)  $- \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i$ 
A)  $- \frac { 9 } { 2 } - \frac { 9 \sqrt { 3 } } { 2 } i$ 
B)  $- \frac { 9 } { 2 } + \frac { 9 \sqrt { 3 } } { 2 } i$ 
C)  $- \frac { 9 \sqrt { 3 } } { 2 } + \frac { 9 } { 2 } \mathrm { i }$ 
D)  $- \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i$

Question 5

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

Accepted Answer

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

Question 6

Find the dot product v · w.
-v = 2i + 3j + k and w = 3i + 7j + 3k

Accepted Answer

A)  -30 
B)  -12 
C)  30 
D)  89 
A)  -30 
B)  -12 
C)  30 
D)  89

Question 7

Find the indicated cross product.
-v = 2i + 2j, w = -3i - 5j + 5k Find v × w.

Accepted Answer

A)  10i - 10j - 4k 
B)  25i - 15j - 11k 
C)  10i - 10j - 16k 
D)  -4i - 10j + 10k 
A)  10i - 10j - 4k 
B)  25i - 15j - 11k 
C)  10i - 10j - 16k 
D)  -4i - 10j + 10k

Question 8

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

Accepted Answer

A)  -15i - 15j + 9k 
B)  -9i + 15j + 15k 
C)  -15i - 15j - 9k 
D)  15j + 9k 
A)  -15i - 15j + 9k 
B)  -9i + 15j + 15k 
C)  -15i - 15j - 9k 
D)  15j + 9k

Question 9

Identify and graph the polar equation.
-$r=3-4 \sin \theta$
 
 A)
 
limacon without inner loop

Accepted Answer

B) 
 
limacon with inner loop 
C) 
 
limacon with inner loop 
D) 
 
limacon without inner loop 
B) 
 
limacon with inner loop 
C) 
 
limacon with inner loop 
D) 
 
limacon without inner loop

Question 10

Write the complex number in rectangular form.
-$9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Accepted Answer

A)  9 
B)  $- 9 \mathrm { i }$ 
C)  $9 \mathrm { i }$ 
D)  $- 9$ 
A)  9 
B)  $- 9 \mathrm { i }$ 
C)  $9 \mathrm { i }$ 
D)  $- 9$

Question 11

Solve the problem.
-If u = 7i - 4j and v = -2i + 9j, find u + v.

Accepted Answer

A)  -9i + 5j 
B)  5i + 5j 
C)  9i + 4j 
D)  4i + 5j 
A)  -9i + 5j 
B)  5i + 5j 
C)  9i + 4j 
D)  4i + 5j

Question 12

Find the unit vector having the same direction as v.
-$$v = - 3 i + 4 j$$

Accepted Answer

A)  $u = - \frac { 5 } { 3 } i + \frac { 5 } { 4 } j$ 
B)  $\mathbf { u } = - \frac { 4 } { 5 } \mathrm { i } + \frac { 3 } { 5 } \mathrm { j }$ 
C)  $\mathbf { u } = - \frac { 3 } { 5 } \mathrm { i } + \frac { 4 } { 5 } \mathrm { j }$ 
D)  $\mathbf { u } = - 15 \mathbf { i } + 20 \mathbf { j }$ 
A)  $u = - \frac { 5 } { 3 } i + \frac { 5 } { 4 } j$ 
B)  $\mathbf { u } = - \frac { 4 } { 5 } \mathrm { i } + \frac { 3 } { 5 } \mathrm { j }$ 
C)  $\mathbf { u } = - \frac { 3 } { 5 } \mathrm { i } + \frac { 4 } { 5 } \mathrm { j }$ 
D)  $\mathbf { u } = - 15 \mathbf { i } + 20 \mathbf { j }$

Question 13

Test the equation for symmetry with respect to the given axis, line, or pole.
-$\mathrm { r } = - 4 \cos \theta$; the polar axis

Accepted Answer

A)  May or may not be symmetric with respect to the polar axis 
B)  Symmetric with respect to the polar axis 
A)  May or may not be symmetric with respect to the polar axis 
B)  Symmetric with respect to the polar axis

Question 14

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 15 , \alpha = 45 ^ { \circ }$$

Accepted Answer

A)  $v = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathbf { i } + \frac { 15 } { 2 } \mathbf { j } \right)$ 
B)  $\mathbf { v } = 15 \left( \frac { 15 } { 2 } \mathbf { i } + \frac { 15 \sqrt { 3 } } { 2 } \mathbf { j } \right)$ 
C)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } \mathrm { i } - \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right\}$ 
D)  $\mathbf { v } = 15 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
A)  $v = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathbf { i } + \frac { 15 } { 2 } \mathbf { j } \right)$ 
B)  $\mathbf { v } = 15 \left( \frac { 15 } { 2 } \mathbf { i } + \frac { 15 \sqrt { 3 } } { 2 } \mathbf { j } \right)$ 
C)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } \mathrm { i } - \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right\}$ 
D)  $\mathbf { v } = 15 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$

Question 15

Find the dot product v · w.
-$$v = - 4 i , \quad w = j$$

Accepted Answer

A)  0 
B)  $- 3$ 
C)  $\sqrt { 17 }$ 
D)  $- 4$ 
A)  0 
B)  $- 3$ 
C)  $\sqrt { 17 }$ 
D)  $- 4$

Question 16

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 14 , \quad \alpha = 0 ^ { \circ }$$

Accepted Answer

A)  $v = 14 i$ 
B)  $v = 14 i + 14 j$ 
C)  $v = 14 j$ 
D)  $v = - 14 j$ 
A)  $v = 14 i$ 
B)  $v = 14 i + 14 j$ 
C)  $v = 14 j$ 
D)  $v = - 14 j$

Question 17

Write the expression in the standard form a + bi.
-$$( - \sqrt { 3 } + i ) ^ { 6 }$$

Accepted Answer

A)  $64 - 64 \sqrt { 3 } i$ 
B)  $- 64$ 
C)  $- 64 \sqrt { 3 } + 64 \mathrm { i }$ 
D)  $64 \mathrm { i }$ 
A)  $64 - 64 \sqrt { 3 } i$ 
B)  $- 64$ 
C)  $- 64 \sqrt { 3 } + 64 \mathrm { i }$ 
D)  $64 \mathrm { i }$

Question 18

Match the graph to one of the polar equations.
-  A) $r = 2 \sin \theta$

Accepted Answer

B)  $r = 2 \cos \theta$ 
C)  $r = 1$ 
D)  $r \sin \theta = 1$ 
B)  $r = 2 \cos \theta$ 
C)  $r = 1$ 
D)  $r \sin \theta = 1$

Question 19

Find the quantity if v = 5i - 7j and w = 3i + 2j.
-$\| \mathbf { v } \| - \| \mathbf { w } \|$

Accepted Answer

A)  $\sqrt { 85 }$ 
B)  $2 \sqrt { 6 } - \sqrt { 13 }$ 
C)  $\sqrt { 74 } - \sqrt { 13 }$ 
D)  $\sqrt { 11 }$ 
A)  $\sqrt { 85 }$ 
B)  $2 \sqrt { 6 } - \sqrt { 13 }$ 
C)  $\sqrt { 74 } - \sqrt { 13 }$ 
D)  $\sqrt { 11 }$

Question 20

Use the vectors in the figure below to graph the following vector.  
-$3 \mathbf { w }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Solve the problem. Leave your answer in polar form. - z=10 3+i3 w=5 1+i1 Find $\frac { Z } { w }$ .

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - $r = \frac { 5 } { 1 + \cos \theta }$

Write the expression in the standard form a + bi. - $\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }$

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

Find the dot product v · w. -v = 2i + 3j + k and w = 3i + 7j + 3k

Find the indicated cross product. -v = 2i + 2j, w = -3i - 5j + 5k Find v × w.

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

Write the complex number in rectangular form. - $9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Solve the problem. -If u = 7i - 4j and v = -2i + 9j, find u + v.

Find the unit vector having the same direction as v. - $v = - 3 i + 4 j$

Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } = - 4 \cos \theta$ ; the polar axis

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis. - $\| \mathbf { v } \| = 15 , \alpha = 45 ^ { \circ }$

Find the dot product v · w. - $v = - 4 i , \quad w = j$

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis. - $\| \mathbf { v } \| = 14 , \quad \alpha = 0 ^ { \circ }$

Write the expression in the standard form a + bi. - $( - \sqrt { 3 } + i ) ^ { 6 }$

Match the graph to one of the polar equations. - $Match the graph to one of the polar equations. - A) r = 2 \sin \theta B) r = 2 \cos \theta C) r = 1 D) r \sin \theta = 1$ A) $r = 2 \sin \theta$ B) $r = 2 \cos \theta$ C) $r = 1$ D) $r \sin \theta = 1$

Find the quantity if v = 5i - 7j and w = 3i + 2j. - $\| \mathbf { v } \| - \| \mathbf { w } \|$

Use the vectors in the figure below to graph the following vector. $Use the vectors in the figure below to graph the following vector. - 3 \mathbf { w }$ - $3 \mathbf { w }$ $Use the vectors in the figure below to graph the following vector. - 3 \mathbf { w }$

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Exam 8: Polar Coordinates; Vectors

Solve the problem. Leave your answer in polar form. - z=10 3+i3 w=5 1+i1 Find Zw\frac { Z } { w }wZ​ .

Plot the point given in polar coordinates. - (−2,9π4)\left( - 2 , \frac { 9 \pi } { 4 } \right)(−2,49π​) A) B) C) D)

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - r=51+cos⁡θr = \frac { 5 } { 1 + \cos \theta }r=1+cosθ5​

Write the expression in the standard form a + bi. - [3(cos⁡5π6+isin⁡5π6)]4\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }[3​(cos65π​+isin65π​)]4

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - r=cos⁡θr = \cos \thetar=cosθ