Question 1

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

Accepted Answer

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

Question 2

Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 4 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Accepted Answer

A)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
A)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$

Question 3

Find the indicated cross product.
-$$\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { w } = - 5 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$$
Find $\mathbf { w } \times \mathbf { v }$.

Accepted Answer

A)  $- 41 \mathrm { i } - 2 \mathrm { j } - 18 \mathrm { k }$ 
B)  $2 \mathbf { i } + 41 \mathbf { j } - 18 \mathbf { k }$ 
C)  $- 2 \mathbf { i } + 40 \mathbf { j } - 18 \mathbf { k }$ 
D)  $- 18 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$ 
A)  $- 41 \mathrm { i } - 2 \mathrm { j } - 18 \mathrm { k }$ 
B)  $2 \mathbf { i } + 41 \mathbf { j } - 18 \mathbf { k }$ 
C)  $- 2 \mathbf { i } + 40 \mathbf { j } - 18 \mathbf { k }$ 
D)  $- 18 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$

Question 4

Write the expression in the standard form a + bi.
-$( \sqrt { 3 } + i ) ^ { 5 }$

Accepted Answer

A)  $- 16 \sqrt { 3 } + 16 \mathrm { i }$ 
B)  $16 - 16 \sqrt { 3 } i$ 
C)  $9 \sqrt { 3 } + 5 i$ 
D)  $16 \sqrt { 3 } - 16 i$ 
A)  $- 16 \sqrt { 3 } + 16 \mathrm { i }$ 
B)  $16 - 16 \sqrt { 3 } i$ 
C)  $9 \sqrt { 3 } + 5 i$ 
D)  $16 \sqrt { 3 } - 16 i$

Question 5

Choose the one alternative that best completes the statement or answers the question.
Find the direction angles of the vector. Round to the nearest degree, if necessary.
-$$\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j } - 12 \mathbf { k }$$

Accepted Answer

A)  $\alpha = 117 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 157 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 107 ^ { \circ } , \gamma = 149 ^ { \circ }$ 
C)  $\alpha = 118 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 161 ^ { \circ }$ 
D)  $\alpha = 92 ^ { \circ } , \beta = 91 ^ { \circ } , \gamma = 94 ^ { \circ }$ 
A)  $\alpha = 117 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 157 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 107 ^ { \circ } , \gamma = 149 ^ { \circ }$ 
C)  $\alpha = 118 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 161 ^ { \circ }$ 
D)  $\alpha = 92 ^ { \circ } , \beta = 91 ^ { \circ } , \gamma = 94 ^ { \circ }$

Question 6

Plot the complex number in the complex plane.
-$-4-6 i$

Accepted Answer

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

Question 7

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
-$$- 15 + 20 \mathrm { i }$$

Accepted Answer

A)  $25 \left( \cos 53.1 ^ { \circ } + i \sin 53.1 ^ { \circ } \right)$ 
B)  $25 \left( \cos 126.9 ^ { \circ } + i \sin 126.9 ^ { \circ } \right)$ 
C)  $25 \left( \cos 306.9 ^ { \circ } + i \sin 306.9 ^ { \circ } \right)$ 
D)  $25 \left( \cos 233.1 ^ { \circ } + \mathrm { i } \sin 233.1 ^ { \circ } \right)$ 
A)  $25 \left( \cos 53.1 ^ { \circ } + i \sin 53.1 ^ { \circ } \right)$ 
B)  $25 \left( \cos 126.9 ^ { \circ } + i \sin 126.9 ^ { \circ } \right)$ 
C)  $25 \left( \cos 306.9 ^ { \circ } + i \sin 306.9 ^ { \circ } \right)$ 
D)  $25 \left( \cos 233.1 ^ { \circ } + \mathrm { i } \sin 233.1 ^ { \circ } \right)$

Question 8

Choose the one alternative that best completes the statement or answers the question.
Find the requested vector.
-$$\mathbf { w } = 3 \mathbf { i } - 3 \mathbf { j } - \mathbf { k }$$
Find a vector orthogonal to both $\mathbf { w }$ and $\mathbf { j } + \mathbf { k }$.

Accepted Answer

A)  $- 4 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }$ 
B)  $4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$ 
C)  $- 2 \mathbf { i } - 3 \mathbf { j } + 3 \mathbf { k }$ 
D)  $3 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k }$ 
A)  $- 4 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }$ 
B)  $4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$ 
C)  $- 2 \mathbf { i } - 3 \mathbf { j } + 3 \mathbf { k }$ 
D)  $3 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k }$

Question 9

Choose the one alternative that best completes the statement or answers the question.
-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 $45 ^ { \circ }$ 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 $2,605.4 \mathrm { lb }$. 
B)  The truck must pull with a force of $4,500.2 \mathrm { lb }$. 
C)  The truck must pull with a force of $4,276.9 \mathrm { lb }$. 
D)  The truck must pull with a force of $2,197.3 \mathrm { lb }$. 
A)  The truck must pull with a force of $2,605.4 \mathrm { lb }$. 
B)  The truck must pull with a force of $4,500.2 \mathrm { lb }$. 
C)  The truck must pull with a force of $4,276.9 \mathrm { lb }$. 
D)  The truck must pull with a force of $2,197.3 \mathrm { lb }$.

Question 10

Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 3 \sin ( 3 \theta ) ;$ the line $\theta = \frac { \pi } { 2 }$

Accepted Answer

A)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
A)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$

Question 11

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
-$$r \sec \theta = - 6$$

Accepted Answer

A) 
  
B) 
 
 $ (x+3)^{2}+y^{2}=9 ; $ circle, radius 3
center $ (-3,0) $ in rectangular coordinates 
C)

$ y=-6 $; horizontal line 6 units below the pole 
D)

$ x=-6 $; vertical line 6 units to the left of the pole

$ x^{2}+(y+3)^{2}=9 ; $ circle, radius 3 , 
center at $ (0,-3) $ in rectangular coordinates 
A) 
  
B) 
 
 $ (x+3)^{2}+y^{2}=9 ; $ circle, radius 3
center $ (-3,0) $ in rectangular coordinates 
C)

$ y=-6 $; horizontal line 6 units below the pole 
D)

$ x=-6 $; vertical line 6 units to the left of the pole

$ x^{2}+(y+3)^{2}=9 ; $ circle, radius 3 , 
center at $ (0,-3) $ in rectangular coordinates

Question 12

Solve the problem.
-If $\mathbf { w } = 3 \mathbf { i } + 3 \mathbf { j }$, find $2 \mathbf { w }$.

Accepted Answer

A)  $6 \mathbf { i } + 6 \mathbf { j }$ 
B)  $6 \mathbf { i } + 3 \mathbf { j }$ 
C)  $5 \mathbf { i } + 3 \mathbf { j }$ 
D)  $5 \mathbf { i } + 5 \mathbf { j }$ 
A)  $6 \mathbf { i } + 6 \mathbf { j }$ 
B)  $6 \mathbf { i } + 3 \mathbf { j }$ 
C)  $5 \mathbf { i } + 3 \mathbf { j }$ 
D)  $5 \mathbf { i } + 5 \mathbf { j }$

Question 13

Write the expression in the standard form a + bi.
-$$( 1 + i ) ^ { 20 }$$

Accepted Answer

A)  $1024 \mathrm { i }$ 
B)  1024 
C)  $- 1024 \mathrm { i }$ 
D)  $- 1024$ 
A)  $1024 \mathrm { i }$ 
B)  1024 
C)  $- 1024 \mathrm { i }$ 
D)  $- 1024$

Question 14

Find the dot product v · w.
-$\mathbf { v } = - 15 \mathrm { i } + 5 \mathbf { j } , \quad \mathbf { w } = 4 \mathbf { i } - 4 \mathrm { j }$

Accepted Answer

A)  $- 60$ 
B)  $- 40$ 
C)  $- 20$ 
D)  $- 80$ 
A)  $- 60$ 
B)  $- 40$ 
C)  $- 20$ 
D)  $- 80$

Question 15

Find the indicated cross product.
-$$\mathbf { v } = - 6 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = - 3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }$$
Find $\mathbf { v } \times \mathbf { w }$.

Accepted Answer

A)  $8 \mathbf { i } - 9 \mathbf { j } + 21 \mathbf { k }$ 
B)  $21 \mathbf { i } - 8 \mathbf { j } + 9 \mathbf { k }$ 
C)  $- 8 \mathbf { i } + 9 \mathbf { j } + 3 \mathbf { k }$ 
D)  $- 2 \mathbf { i } - 3 \mathbf { j } + 39 \mathbf { k }$ 
A)  $8 \mathbf { i } - 9 \mathbf { j } + 21 \mathbf { k }$ 
B)  $21 \mathbf { i } - 8 \mathbf { j } + 9 \mathbf { k }$ 
C)  $- 8 \mathbf { i } + 9 \mathbf { j } + 3 \mathbf { k }$ 
D)  $- 2 \mathbf { i } - 3 \mathbf { j } + 39 \mathbf { k }$

Question 16

Find v .
-$$\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }$$

Accepted Answer

A)  $\sqrt { 97 }$ 
B)  5 
C)  $\sqrt { 61 }$ 
D)  $\sqrt { 77 }$ 
A)  $\sqrt { 97 }$ 
B)  5 
C)  $\sqrt { 61 }$ 
D)  $\sqrt { 77 }$

Question 17

Find the indicated cross product.
-$$v = i + 2 j + 2 k , \quad w = 6 i + 5 j - k$$
Find $\mathbf { v } \times \mathbf { w }$.

Accepted Answer

A)  $- 12 i + 13 j - 7 k$ 
B)  $- 7 \mathbf { i } + 12 \mathbf { j } - 13 \mathbf { k }$ 
C)  $9 \mathbf { i } - 8 \mathbf { j } - 28 \mathbf { k }$ 
D)  $8 \mathrm { i } - 11 \mathrm { j } + 17 \mathbf { k }$ 
A)  $- 12 i + 13 j - 7 k$ 
B)  $- 7 \mathbf { i } + 12 \mathbf { j } - 13 \mathbf { k }$ 
C)  $9 \mathbf { i } - 8 \mathbf { j } - 28 \mathbf { k }$ 
D)  $8 \mathrm { i } - 11 \mathrm { j } + 17 \mathbf { k }$

Question 18

Write the complex number in rectangular form.
-$6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Accepted Answer

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

Question 19

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
-$$- 4 + 4 \mathrm { i }$$

Accepted Answer

A)  $4 \sqrt { 2 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ 
B)  $4 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $16 \left( \cos 135 ^ { \circ } + \mathrm { i } \sin 135 ^ { \circ } \right)$ 
D)  $16 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right)$ 
A)  $4 \sqrt { 2 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ 
B)  $4 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $16 \left( \cos 135 ^ { \circ } + \mathrm { i } \sin 135 ^ { \circ } \right)$ 
D)  $16 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right)$

Question 20

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

Accepted Answer

A)  $2 \sqrt { 6 } + \sqrt { 13 }$ 
B)  $\sqrt { 74 } + \sqrt { 13 }$ 
C)  $\sqrt { 39 }$ 
D)  $\sqrt { 87 }$ 
A)  $2 \sqrt { 6 } + \sqrt { 13 }$ 
B)  $\sqrt { 74 } + \sqrt { 13 }$ 
C)  $\sqrt { 39 }$ 
D)  $\sqrt { 87 }$

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

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 4 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Find the indicated cross product. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { w } = - 5 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ Find $\mathbf { w } \times \mathbf { v }$ .

Write the expression in the standard form a + bi. - $( \sqrt { 3 } + i ) ^ { 5 }$

Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j } - 12 \mathbf { k }$

Plot the complex number in the complex plane. - $-4-6 i$

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 15 + 20 \mathrm { i }$

Choose the one alternative that best completes the statement or answers the question. Find the requested vector. - $\mathbf { w } = 3 \mathbf { i } - 3 \mathbf { j } - \mathbf { k }$ Find a vector orthogonal to both $\mathbf { w }$ and $\mathbf { j } + \mathbf { k }$ .

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 3 \sin ( 3 \theta ) ;$ the line $\theta = \frac { \pi } { 2 }$

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - $r \sec \theta = - 6$ $Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - r \sec \theta = - 6$

Solve the problem. -If $\mathbf { w } = 3 \mathbf { i } + 3 \mathbf { j }$ , find $2 \mathbf { w }$ .

Write the expression in the standard form a + bi. - $( 1 + i ) ^ { 20 }$

Find the dot product v · w. - $\mathbf { v } = - 15 \mathrm { i } + 5 \mathbf { j } , \quad \mathbf { w } = 4 \mathbf { i } - 4 \mathrm { j }$

Find the indicated cross product. - $\mathbf { v } = - 6 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = - 3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }$ Find $\mathbf { v } \times \mathbf { w }$ .

Find v . - $\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }$

Find the indicated cross product. - $v = i + 2 j + 2 k , \quad w = 6 i + 5 j - k$ Find $\mathbf { v } \times \mathbf { w }$ .

Write the complex number in rectangular form. - $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 4 + 4 \mathrm { i }$

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

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Exam 8: Polar Coordinates; Vectors

Test the equation for symmetry with respect to the given axis, line, or pole. - r=4−4sin⁡θ\mathrm { r } = 4 - 4 \sin \thetar=4−4sinθ ; polar axis

Test the equation for symmetry with respect to the given axis, line, or pole. - r=4cos⁡θ;r = 4 \cos \theta ;r=4cosθ; the line θ=π2\theta = \frac { \pi } { 2 }θ=2π​

Find the indicated cross product. - v=4i+2j+5k,w=−5i+2j+4k\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { w } = - 5 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }v=4i+2j+5k,w=−5i+2j+4k Find w×v\mathbf { w } \times \mathbf { v }w×v .

Write the expression in the standard form a + bi. - (3+i)5( \sqrt { 3 } + i ) ^ { 5 }(3​+i)5

Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - v=−6i−4j−12k\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j } - 12 \mathbf { k }v=−6i−4j−12k

Plot the complex number in the complex plane. - −4−6i-4-6 i−4−6i

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - −15+20i- 15 + 20 \mathrm { i }−15+20i

Choose the one alternative that best completes the statement or answers the question. Find the requested vector. - w=3i−3j−k\mathbf { w } = 3 \mathbf { i } - 3 \mathbf { j } - \mathbf { k }w=3i−3j−k Find a vector orthogonal to both w\mathbf { w }w and j+k\mathbf { j } + \mathbf { k }j+k .

Test the equation for symmetry with respect to the given axis, line, or pole. - r=3sin⁡(3θ);r = 3 \sin ( 3 \theta ) ;r=3sin(3θ); the line θ=π2\theta = \frac { \pi } { 2 }θ=2π​

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - rsec⁡θ=−6r \sec \theta = - 6rsecθ=−6

Solve the problem. -If w=3i+3j\mathbf { w } = 3 \mathbf { i } + 3 \mathbf { j }w=3i+3j , find 2w2 \mathbf { w }2w .

Write the expression in the standard form a + bi. - (1+i)20( 1 + i ) ^ { 20 }(1+i)20

Find the dot product v · w. - v=−15i+5j,w=4i−4j\mathbf { v } = - 15 \mathrm { i } + 5 \mathbf { j } , \quad \mathbf { w } = 4 \mathbf { i } - 4 \mathrm { j }v=−15i+5j,w=4i−4j

Find the indicated cross product. - v=−6i−3j+k,w=−3i−5j−k\mathbf { v } = - 6 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = - 3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }v=−6i−3j+k,w=−3i−5j−k Find v×w\mathbf { v } \times \mathbf { w }v×w .

Find v . - v=6i+5j−6k\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }v=6i+5j−6k

Find the indicated cross product. - v=i+2j+2k,w=6i+5j−kv = i + 2 j + 2 k , \quad w = 6 i + 5 j - kv=i+2j+2k,w=6i+5j−k Find v×w\mathbf { v } \times \mathbf { w }v×w .

Write the complex number in rectangular form. - 6(cos⁡330∘+isin⁡330∘)6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)6(cos330∘+isin330∘)

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - −4+4i- 4 + 4 \mathrm { i }−4+4i

Find the quantity if v = 5i - 7j and w = 3i + 2j. - ∥v∥+∥w∥\| v \| + \| w \|∥v∥+∥w∥