Deck 9: Vectors in Two and Three Dimensions

Question 1

Use DeMoivre's Theorem to find the indicated power.

Answer 1

Question 2

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Answer 2

$r = 4$

Question 3

Convert the polar equation to rectangular coordinates. $r + \cos \theta = 3$

Answer 3

$\left( x ^ { 2 } + y ^ { 2 } + x \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

Question 4

Graph the polar equation $r = 8 \cos \theta$ .

Question 5

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r = 5 \sec \theta$

Question 6

Convert the point whose polar coordinates are $( 8,5 \pi / 4 )$ to rectangular coordinates.

Question 7

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.]

Question 8

Convert the equation to polar form.

Question 9

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - 2 \sqrt { 3 } , - 2 )$

Question 10

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = \cos 2 \theta$

Question 11

Write $z _ { 1 } = 1 - i$ in polar form then find $1 / z _ { 1 }$

Question 12

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and the other with $r < 0$

Question 13

Convert the polar equation to rectangular coordinates. $\frac {r } { 6 } = \sec \theta$

Question 14

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( 0 , - \sqrt { 5 } )$

Question 15

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 \sqrt { 3 } - 5 i$

Question 16

Let $z _ { 1 } = 2 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ . Find $Z _ { 1 } Z _ { 2 }$

Question 17

Write the complex number in polar form.

Question 18

Sketch a graph of the polar equation. $r = - 3 \cos 3 \theta$

Question 19

Let $z _ { 1 } = \sqrt { 2 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right)$ and $z _ { 2 } = 2 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$ . Find $z _ { 1 } / z _ { 2 }$

Question 20

Find the modulus and the argument for the complex number. $z = - 10 i$

Question 21

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Question 22

Find the cube roots of $i$

Question 23

If a projectile is fired with an initial speed of $v _ { 0 }$ ft/s at an angle $\alpha$
above the horizontal, then its position after t seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t \text { and } y = \left( v _ { 0 } \sin \alpha \right) t - 16 t ^ { 2 }$
where x and y are measured in feet.Suppose a gun fires a bullet into the air with an initial speed of 1024 ft/s at an angle of $30 ^ { \circ }$ to the horizontal. What is the maximum height attained by the bullet?

Question 24

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

Question 25

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r = 3 \cos \theta \csc \theta$

Question 26

Convert the point whose polar coordinates are
to rectangular coordinates.

Question 27

Find the modulus and the argument for the complex number. $z = - i$

Question 28

Graph the polar equation $r= 4 \sin \theta$

Question 29

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.]

Question 30

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

Question 31

Convert the polar equation to rectangular coordinates. $r + \cos \theta = 4$

Question 32

Sketch the curve represented by the parametric equations and find its rectangular-coordinate equation.

Question 33

Convert the polar equation to rectangular coordinates.

Question 34

Sketch a graph of the polar equation. $r = - 2 \sin 3 \theta$

Question 35

Find parametric equations for the line with the given properties.Passing through
and the origin

Question 36

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = 9 \cos 2 \theta$

Question 37

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

Question 38

Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Question 39

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( 0 , - \sqrt { 2 } )$

Question 40

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r = 3 \cos \theta \csc \theta$
I $\text { symmetric about the polar axis }$
II $\text { symmetric about the pole }$
III $\text { symmetric about the line } \theta = \pi / 2$

A)I only
B)I and II
C) I and III
D)II and III
E) none of these

Question 41

Which of the following is not a polar point representation for the point ?

A)
B)
C)
D)
E)

Question 42

Sketch the curve represented by the parametric equations and find its rectangular-coordinate equation.

Question 43

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 \sqrt { 3 } + 5 i$

Question 44

Convert the polar equation to rectangular coordinates. $r + \cos \theta = 4$

A) $\left( x ^ { 2 } + y ^ { 2 } + x \right) ^ { 2 } = 16 \left( x ^ { 2 } + y ^ { 2 } \right)$
B) $x ^ { 2 } + y ^ { 2 } = 16 ( x + y )$
C) $\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 16 \left( x ^ { 2 } - y ^ { 2 } \right)$
D) $\left( x ^ { 2 } + y ^ { 2 } + 2 x \right) ^ { 2 } = 4 ( x + y )$
E) $\left( x ^ { 2 } + y ^ { 2 } + x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)$

Question 45

Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Question 46

Solve the equation.

Question 47

Write $z _ { 1 } = 1 - \sqrt { 2 j }$ in polar form then find $1 / z _ { 1 }$

Question 48

Let $z _ { 1 } = 4 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ . Find $Z _ { 1 } Z _ { 2 }$

Question 49

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 9$

A) $r ^ { 2 } = 9 \csc 2 \theta$
B) $r = 3 \sec 2 \theta$
C) $r ^ { 2 } = 9 \sec 2 \theta$
D) $r ^ { 2 } = 9 \csc \theta$
E)none of these

Question 50

Convert the rectangular coordinates to polar coordinates with

A)
B)
C)
D)
E)none of these

Question 51

Convert the polar equation to rectangular coordinates. $\frac {r } { 6 } = \sec \theta$

A) $y = 6$
B) $y = 6 x$
C) $x = 6$
D) $x y = 6$
E)none of these

Question 52

Graph the polar equation $r = 8 \cos \theta$

A)

B)

C)

D)

Question 53

Convert the point whose polar coordinates are $( \sqrt { 3 } , \pi / 6 )$ to rectangular coordinates

A) $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$

B) $\left( \frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$

C) $( 3 , \sqrt { 3 } )$

D) $( \sqrt { 3 } , \pi / 6 )$

E) $\left( \frac { 2 } { 3 } , \frac { \sqrt { 2 } } { 2 } \right)$

Question 54

Convert the rectangular coordinates to polar coordinates with

A)
B)
C)
D)
E)

Question 55

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

A) $r^ { 2 } = 4$
B) $4 r= \cos \theta + \sin \theta$
C) $y = 4 \cos \theta + 4 \sin \theta$
D) $r = 4$
E)none of these

Question 56

Find parametric equations for the line with the given properties.Passing through
and the origin

Question 57

Use DeMoivre's Theorem to find the indicated power. $( 1 - \sqrt { 3 } i ) ^ { 5 }$

Question 58

Let $z _ { 1 } = 8 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$ and $z _ { 2 } = 2 \sqrt { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ ) Find $z _ { 1 } / z _ { 2 }$

A) $3 - 4 i$
B) $4 - \sqrt { 3 } i$
C) $4 + \sqrt { 3 } i$
D) $\frac { 4 } { 3 }$
E) $- \frac { 4 \sqrt { 3 } } { 3 } i$

Question 59

Which of the following is not a polar point representation for the point ?

A)
B)
C)
D)
E)

Question 60

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$$z = 5 - 5 i \sqrt { 3 }$

A) $10 ( \cos ( \pi / 3 ) + i \sin ( \pi / 3 ) )$

B) $5 ( \cos ( 5 \pi / 3 ) + i \sin ( 5 \pi / 3 ) )$

C) $10 ( \cos ( \pi / 6 ) + i \sin ( \pi / 6 ) )$

D) $5 ( \cos ( 7 \pi / 6 ) + i \sin ( 7 \pi / 6 ) )$

E) none

Question 61

Sketch a graph of the polar equation. $r = \sqrt { 3 } - \cos \theta$

A)

B)

C)

D)

Question 62

Find the modulus and the argument for the complex number. $z = - i$

A) $r = - 1 , \theta = \frac { \pi } { 2 }$
B) $r = 1 , \theta = \pi$
C) $r = i , \theta = 0$
D) $r = \sqrt { 2 } , \theta = \frac { 3 \pi } { 2 }$
E) $r = 1 , \theta = \frac { 3 \pi } { 2 }$

Question 63

Let $z _ { 1 } = 4 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ ) Find $Z _ { 1 } Z _ { 2 }$

A) $20 + i$
B) $20 - 20 i$
C) $20$
D) $0 - 20 i$
E)none

Question 64

Find the rectangular-coordinate equation for the parametric equations given.

Question 65

Convert the point whose polar coordinates are to rectangular coordinates

A)
B)
C)
D)
E)

Question 66

Write the complex number in polar form. $z = - 1 - \sqrt { 3 } i$

A) $z = 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

B) $z = 2 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$

C) $z = 2 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$

D) $z = 2 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$

E) $z = 2 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$

Question 67

Solve the equation.

A)
B)
C)
D)
E)

Question 68

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = \cos 2 \theta$
I $\text { symmetric about the polar axis }$
II $\text { symmetric about the pole }$
III $\text { symmetric about the line } \theta = \pi / 2$

A)I only
B)I and II
C) I and III
D)II and III
E) I, II, III

Question 69

Graph the polar equation $r = 8 \cos \theta$

A)

B)

C)

D)

Question 70

Use DeMoivre's Theorem to find the indicated power.

A)
B)
C)
D)
E)

Question 71

Find parametric equations for the line with the given properties.Passing through and the origin

A)
B)
C)
D)
E)none of these

Question 72

Write in polar form then find

A)
B)
C)
D)
E)

Question 73

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

A)

B)

C)

D)

Question 74

Find a rectangular-coordinate equation for the curve by eliminating the parameter.

A)
B)
C)
D)
E) none of these

Question 75

If a projectile is fired with an initial speed of ft/s at an angle
Above the horizontal, then its position after t seconds is given by the parametric equations
Where x and y are measured in feet. Suppose a gun fires a bullet into the air with an initial speed of 1024 ft/s at an angle of
To the horizontal. How far from the gun will the bullet hit the ground?

A)
B)
C)
D)
E)

Question 76

Convert the rectangular coordinates to polar coordinates with

A)
B)
C)
D)
E)

Question 77

Sketch a graph of the polar equation. $r=\sqrt{3}-\cos \theta$

A)

B)

C)

D)

Question 78

Let $z _ { 1 } = \sqrt { 2 } \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ and $z _ { 2 } = \sqrt { 2 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ ) Find $Z _ { 1 } Z _ { 2 }$

A) $\sqrt { 2 } + i$
B) $2 - 2 i$
C) $\frac { \sqrt { 2 } } { 2 }$
D) $2 j$
E)none

Question 79

Convert the polar equation to rectangular coordinates.

A)
B)
C)
D)
E)none

Question 80

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 2 , y = \frac { t } { t + 2 }$

A) $y = \frac { x - 4 } { x }$

B) $y = \frac { x - 2 } { x }$

C) $y = \frac { x - 2 } { 4 x }$

D) $y = \frac { 2 x - 1 } { 2 x }$

E) none of these