Deck 8: Polar Coordinates, Vectors, and Complex Numbers

Question 1

Evaluate $|24-10 i|$ .

Answer 1

26

Question 2

Find two real numbers b such that $|15+b i|$ = 17.

Answer 2

8 or -8

Question 3

Find two real numbers a such that $|a+4 i|$ = 5.

Answer 3

3 or -3

Question 4

Write $7-7 i$ in polar form. Give the exact answer using radians for $\theta$ .

Question 5

Write $-11+11 \sqrt{3} i$ in polar form. Give the exact answer using radians for $\theta$ .

Question 6

Write $\frac{1}{11\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)}$ in polar form. Give the exact answer using radians for $\theta$ .

Question 7

Write $\left(\cos \frac{\pi}{14}+i \sin \frac{\pi}{14}\right)\left(\cos \frac{\pi}{15}+i \sin \frac{\pi}{15}\right)$ in polar form. Give the exact answer using radians for $\theta$ .

Question 8

Evaluate $(2-2 i)^{277}$ .

Question 9

Write 7 in polar form.

Question 10

Write 10i in polar form. Give the exact answer using radians for $\theta$ .

Question 11

Suppose $r_{1}=10\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)$ and $r_{2}=5\left(\cos \frac{\pi}{15}+i \sin \frac{\pi}{15}\right)$ . Find the polar form of $r_{1} r_{2}$ . Give the exact answer using radians for $\theta$ .

Question 12

Suppose $r_{1}=12\left(\cos \frac{7 \pi}{11}+i \sin \frac{7 \pi}{11}\right)$ and $r_{2}=3\left(\cos \frac{\pi}{11}+i \sin \frac{\pi}{11}\right)$ . Find the polar form of $\frac{r_{1}}{r_{2}}$ . Give the exact answer using radians for $\theta$ .

Question 13

Find four distinct numbers z such that $z^{4}=-4$ . Give the exact answer.

Question 14

Evaluate $(\sqrt{3}-i)^{424}$ .

Question 15

Find three distinct numbers z such that $z^{3}=1$ . Give the exact answer.

Question 16

Evaluate $|5 \cos \theta+5 i \sin \theta|$ .

Question 17

Evaluate $(8+15 i)(8-15 i)$ .

Question 18

Find the complex conjugate of $(12+5 i)$ .

Question 19

Evaluate $|-9-5 i|$ . Give the exact answer.

Question 20

Write $2+2 i$ in polar form. Give the exact answer using radians for $\theta$ .

Question 21

Write $5 \sqrt{3}+5 i$ in polar form. Give the exact answer using radians for $\theta$ .

Question 22

Write $-4-4 \sqrt{3} i$ in polar form. Give the exact answer using radians for $\theta$ .

Question 23

Evaluate $(4-4 i)^{271}$

Question 24

Evaluate $(\sqrt{3}-i)^{403}$ .

A) $2^{403}(-\sqrt{3}+i)$
B) $2^{402}(-\sqrt{3}+i)$
C) $2^{403}(\sqrt{3}-i)$
D) $2^{402}(\sqrt{3}-i)$

Question 25

Find four distinct complex numbers z such that z⁴ = -5. Give the exact answers.

Question 26

Find three distinct complex numbers z such that z³ = -2

A) $\frac{\sqrt[3]{2}}{2}(1+\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(1-\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(-1+\sqrt{3} i)$
B) $\sqrt[3]{2}, \frac{\sqrt[3]{2}}{2}(1+\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(1-\sqrt{3} i)$
C) $-\sqrt[3]{2}, \frac{\sqrt[3]{2}}{2}(1+\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(1-\sqrt{3} i)$
D) $\frac{\sqrt[3]{2}}{2}(1+\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(1-\sqrt{3} i), \frac{\sqrt[3]{2}}{2}(-1-\sqrt{3} i)$

Question 27

The complex number $z = 4^{\frac{1}{6}} \left(-\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$ satisfies the equation $z^6$ = -4

Question 28

Which of the following complex numbers satisfy the equation z³ = 7i?

A) $z=\sqrt[3]{7}\left(-\frac{\sqrt{3}}{2}+\frac{i}{2}\right)$
B) $z=\sqrt[3]{7}\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)$
C) $z=\sqrt[3]{7}\left(\frac{1}{2}-\frac{\sqrt{3} i}{2}\right)$
D) $z=\sqrt[3]{7}\left(-\frac{1}{2}+\frac{\sqrt{3} i}{2}\right)$

Question 29

Write the expression in the form a + bi, where a and b are real numbers.
(2 + 10i) + (9 + 5i)

Question 30

Write the expression in the form a + bi, where a and b are real numbers.
(8 + 7i) - (4 + 10i)

Question 31

Write the expression in the form a + bi, where a and b are real numbers.
(3 + 6i) - (2 - 5i)

Question 32

Write the expression in the form a + bi, where a and b are real numbers.
(2 + 7i)(9 + 9i)

Question 33

Write the expression in the form a + bi, where a and b are real numbers.
(10 + 4i)(6 - 4i)

Question 34

Write the expression in the form a + bi, where a and b are real numbers.
(4 - 4i)(10 - 9i)

Question 35

Write the expression in the form a + bi, where a and b are real numbers.
(8 + 9i)²

Question 36

Write the expression in the form a + bi, where a and b are real numbers.
(5 - 9i)²

Question 37

Write the expression in the form a + bi, where a and b are real numbers. $(1+\sqrt{3} i)^{2}$

Question 38

Write the expression in the form a + bi, where a and b are real numbers. $(9-\sqrt{7} i)^{2}$

Question 39

Write the expression in the form a + bi, where a and b are real numbers. $(\sqrt{11}-\sqrt{7 i} i)^{2}$

Question 40

Write the expression in the form a + bi, where a and b are real numbers.
(7 + 6i)³

Question 41

Write the expression in the form a + bi, where a and b are real numbers.
$\left(\frac{1}{4}+\frac{\sqrt{15}}{4} i\right)^{2}$

Question 42

Write the expression in the form a + bi, where a and b are real numbers.
i¹¹⁸²

Question 43

Write the expression in the form a + bi, where a and b are real numbers.
$\overline{9+4 i}$

Question 44

Write the expression in the form a + bi, where a and b are real numbers.
$\frac{5+8 i}{9+7 i}$

Question 45

Write the expression in the form a + bi, where a and b are real numbers.
$\frac{1+4 i}{1-9 i}$

Question 46

Suppose w and z are complex numbers. If the real part of wz equals the real part of w times the real part of z, then either w or z is a real number.

Question 47

If z is any complex number, then $z \neq \overline{z}$

Question 48

Let z = a + bi be any complex number such that $z \neq 0$ , then $\frac{1}{z} = \frac{a-bi}{a^2+b^2}$

Question 49

Write the expression in the form a + bi, where a and b are real numbers.
(4 + 9i) - (3 - 10i)

A) 1 + 19i
B) -1 + 19i
C) 1 - 19i
D) -1 - 19i

Question 50

Write the expression in the form a + bi, where a and b are real numbers.
(-7 + 3i)(5 - 2i)

A) $-29-29 i$
B) $-29+29 i$
C) $29+29 i$
D) $29-29 i$

Question 51

Which of the following alternatives corresponds to (a + bi)² ?

A) a² + b²
B) a² + b²i
C) (a² + b²) + (2ab)i
D) (a² - b²) + (2ab)i

Question 52

Write the expression in the form a + bi, where a and b are real numbers. $\frac{5+13 i}{3-2 i}$

A) $\frac{11}{5}+\frac{49}{5} i$
B) $-\frac{11}{5}+\frac{49}{5} i$
C) $-\frac{11}{13}+\frac{49}{13} i$
D) $-\frac{11}{13}-\frac{49}{13} i$

Question 53

Find two complex numbers whose sum equals 14 and whose product equals 58.

Question 54

Find two possible complex solution to the equation. Give the exact answers.
$12 y^{2}-16 y-5=0$

Question 55

Suppose u = (8, -1). Evaluate $|\mathbf{u}|$

A) $\sqrt{65}$
B) 65
C) 7
D) 63

Question 56

Find two distinct numbers k such that $|k(5,3)|=23$ . Give the exact answers.

Question 57

Suppose u = (-1, 4) and v = (3, -1).

A) Draw a figure illustrating the sum of u and v as arrows.
B) Compute the sum u + v using coordinates.

Question 58

Suppose u = (-3, 4) and v = (5, -2).

A) Draw a figure illustrating the difference of u and v as arrows.
B) Compute the sum u - v using coordinates.

Question 59

Suppose u = (2, 1) and v = (3, 1). Compute $\mathrm{u} \cdot \mathrm{v}$ .

Question 60

Use the dot product to find the angle between the vectors (-2, -4) and (4, 5). Express your answer in radians to four decimal places.

Question 61

If u and v are two vectors with the same initial point, then $\left| u-v \right|$ equals the distance between the endpoint of u and the endpoint of v.

Question 62

Let u and v be vectors. Determine which of the following alternatives is always true.

A) $|\mathbf{u} \cdot \mathbf{v}| \geq|\mathbf{u}||\mathbf{v}|$
B) $|\mathbf{u}+\mathbf{v}|^{2}=|\mathbf{u}|^{2}+|\mathbf{v}|^{2}$
C) $|\mathbf{u}|^{2}+|\mathbf{v}|^{2}=|\mathbf{u}+\mathbf{v}|^{2}-|\mathbf{u}-\mathbf{v}|^{2}$
D) $|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|$

Question 63

Suppose v = (a , b) and $|\mathrm{v}|=5$ . Which of the following vectors have magnitude 5?

A) (b, a)
B) (-b, -a)
C) (-a, -b)
D) All of them

Question 64

Find a complex number z such that $\bar{z}=\frac{1}{z}$

A) $\frac{1}{2}+\frac{i}{2}$
B) $-\frac{3}{4}-\frac{i}{4}$
C) $\frac{3}{5}+\frac{4 i}{5}$
D) $\frac{1}{3}+\frac{2 i}{3}$

Question 65

Let $z=4\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)$ Be a complex number. Find $\frac{1}{z}$ .

A) $\frac{1}{4}\left(\cos \frac{\pi}{7}-i \sin \frac{\pi}{7}\right)$
B) $\frac{1}{4}\left(-\cos \frac{\pi}{7}-i \sin \frac{\pi}{7}\right)$
C) $\frac{1}{4}\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)$
D) $\frac{1}{4}\left(-\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)$

Question 66

Let z = 10-2i and w = 9-2i be two complex numbers. The imaginary part of z + w is -4i.

Question 67

Find an angle that determines the direction of the vector (-2, 5). Express your answer in decimal radians rounded to three decimal places.

Question 68

Find an angle that determines the direction of the vector (-6, -4). Express your answer in decimal radians rounded to three decimal places.

Question 69

Suppose the wind at airplane heights is 35 miles per hour (relative to the ground) moving 25° north of east. Relative to the wind, an airplane is flying at 350 miles per hour 55° south of the wind. Find the speed and direction of the airplane relative to the ground. Round the speed to 2 decimal places and the angle to the tenth of a degree.

Question 70

Suppose the wind at airplane heights is 40 miles per hour (relative to the ground) moving 18° south of east. An airplane wants to fly directly north at 340 miles per hour relative to the ground. Find the speed and direction that the airplane must fly relative to the wind. Round the speed to 2 decimal places and the direction to a tenth of a degree.

Question 71

Use the dot product to find the angle between the vectors (3, 9) and (2, 5). Express your answer in radians to four decimal places.

Question 72

Suppose u = (3, -3) and v = (-2, 3). Find u + v.

Question 73

Suppose u = (3, -5) and v = (4, -3). Find u - v.

Question 74

Suppose u = (3, -6). Find -7u.

Question 75

Convert the polar coordinates r = 6, $\theta=8 \pi$
To rectangular coordinates in the xy-plane.

A) (0, 6)
B) (0, -6)
C) (6, 0)
D) (-6, 0)

Question 76

Convert the polar coordinates r = 6, $\theta=2^{500} \pi$ To rectangular coordinates in the xy-plane.

A) (6, 0)
B) (-6, 0)
C) (0, 6)
D) (0, -6)

Question 77

Convert the polar coordinates r = 5, $\theta=\frac{15 \pi}{2}$ to rectangular coordinates in the xy-plane.

Question 78

Convert the polar coordinates r = 5, $\theta=\frac{\pi}{6}$ to rectangular coordinates in the xy-plane. Give the exact answer.

Question 79

Convert the polar coordinates r = 17, $\theta=\frac{11 \pi}{4}$ to rectangular coordinates in the xy-plane. Give the exact answer.

Question 80

Convert the polar coordinates r = 3, $\theta=\frac{7 \pi}{3}$ to rectangular coordinates in the xy-plane. Give the exact answer.