a. describe vector...1. sec 7.1 – basic vector forms name: 1. describe each of the following...

1. Sec 7.1 – Basic Vector Forms Name:

1. Describe each of the following vectors in rectangular form ⟨𝑥, 𝑦⟩ and using component vectors i and j (e.g. 3i + 4j)

A. Describe Vector 𝐴𝐵⃗⃗⃗⃗ ⃗:

B. Describe Vector 𝐶𝐷⃗⃗⃗⃗ ⃗:

C. Describe Vector 𝐸𝐹⃗⃗⃗⃗ ⃗:

D. Describe Vector 𝐺𝐻⃗⃗⃗⃗⃗⃗ :

2. Describe each of the following vectors in rectangular form ⟨𝑥, 𝑦⟩ and using component vectors i and j (e.g. 3i + 4j)

A. Describe Vector 𝑀𝑁⃗⃗⃗⃗⃗⃗ ⃗ given

point M(– 4, 2) and N(2, 6)

B. Describe Vector 𝑆𝑇⃗⃗ ⃗⃗ given

point S(3, – 2) and T (– 4, 6)

3. Create a graph of each of the following vectors in standard form.

A. The vector 𝑣 : ⟨−3,1⟩

B. The vector �⃗⃗� : 4𝒊 − 2𝒋

C. The vector 𝐴𝐵⃗⃗⃗⃗ ⃗ given point

A(–2, 2) and B(1, 6)

M. Winking © Unit 7-1 pg. 123

4. Describe each of the following vectors in polar form ⟨𝑟, 𝜗⟩ . A. Describe Vector 𝑝 :

B. Describe Vector 𝑞 :

C. Describe Vector �⃗⃗� :

5. Create a graph of each of the following vectors in standard form.

A. The vector 𝑣 : ⟨4,150°⟩

B. The vector �⃗� : ⟨3, −140°⟩

C. The vector 𝑝 : ⟨−2,60°⟩

6. Determine at least 4 ways to describe vector �⃗⃗� in polar form:


7. Rewrite each of the following vectors from polar form to rectangular form. A. Rewrite vector �⃗⃗� in rectangular form and graph it on the rectangular graph paper in standard position.

B. Rewrite vector �⃗� in rectangular form and graph it on the rectangular graph paper in standard position.

C. Rewrite vector 𝑝 in rectangular form and graph it on the rectangular graph paper in standard position.

8. Consider the golfer below. He struck the ball so that it was moving at a speed of 200 feet per second at a 70° angle.

A. Write the vector in polar form.

B. Write the vector in rectangular form.

C. If we assumed that the green was 800 feet away horizontally, how long would it

take the golf ball to reach the green (assuming air resistance wasn’t a factor)?

70°


9. Consider the ramp shown below. A car that weighs 2300 pounds is being pushed up a ramp with a 9° elevation.

How many pounds of force must the people use just to hold

the car in place on the ramp?

10. Rewrite each of the following vectors from rectangular form to polar form.

A. Rewrite vector 𝐴𝐵⃗⃗⃗⃗ ⃗ in polar form and graph it on the polar graph paper in standard position.

B. Rewrite vector �⃗� in polar form and graph it on the polar graph paper in standard position.

C. Rewrite vector 𝐶𝐷⃗⃗⃗⃗ ⃗ in polar form and graph it on the polar graph paper in standard position.


Finding Theta (where α is the reference angle)

Quad 1 (x & y positive) e.g. �⃗⃗� : ⟨𝟑, 𝟐⟩

Quad 2 (x negative, y positive) e.g. �⃗� : ⟨−𝟐, 𝟐⟩

Quad 3 (x & y negative) e.g. 𝑝 : ⟨−𝟐,−𝟑⟩

Quad 4 (x negative, y positive) e.g. 𝑞 : ⟨𝟏, −𝟑⟩

𝒕𝒂𝒏−𝟏 (|𝟐|

|𝟑|) = 𝜶 ≈ 𝟑𝟑. 𝟕°

𝜽 ≈ 𝟑𝟑. 𝟕°

𝒕𝒂𝒏−𝟏 (|2|

|−2|) = 𝛼 = 45°

𝜃 = 180° − 45° = 𝟏𝟑𝟓°

𝒕𝒂𝒏−𝟏 (|−𝟑|

|−𝟐|) = 𝜶 ≈ 𝟓𝟔. 𝟑°

𝜽 = 180° + 56.3° ≈ 𝟐𝟑𝟔. 𝟑°

𝒕𝒂𝒏−𝟏 (|−𝟑|

|𝟏|) = 𝜶 ≈ 𝟕𝟏. 𝟔°

𝜽 = 360° − 71.6° ≈ 𝟐𝟖𝟖. 𝟒

= α = 180° – α = 180° + α = 360° – α

11. Perform the following vector operations. A. Consider �⃗⃗� : ⟨−7,4⟩ determine ‖�⃗⃗� ‖. B. Rewrite vector �⃗� : ⟨9 , 260°⟩ in rectangular form.

12. A balloon is floating up in to the sky at a rate of 6 feet per second. At the same time the wind is blowing the balloon

horizontally due east at a speed of 10 feet per second. A. How fast is the balloon actually moving?

B. What is the angle of elevation of the balloon’s ascent?

13. A barge full of containers is being moved across a bay. One tow boat is pulling the barge due North with a force of 5000 Newtons. A second tow boat is pulling the barge due West with a force of 3800 Newtons.

A. In what direction will the barge move?

B. How much force is being applied in that directions?

6 f

ps


1. Sec 7.2 – Basic Vector Operations Name:

1. Create the following vector statement in the graph to determine the resultant vector in rectangular form.

A. Using the information below graphically determine �⃗⃗� + 𝒋 + 𝟐�⃗⃗⃗�

B. Algebraically find 2p j m

C. Using the information below graphically determine �⃗⃗� − 𝟑�⃗⃗� + 𝟐�⃗⃗�

D. Algebraically find 3 2k q n

Show work

here

:

:

2 :

p

j

m

1b.

1a.

Show work

here

:

3 :

2 :

k

q

n

1d.

Thes

e sh

ou

ld

be

the

sam

e

1c.

The

se s

houl

d

be th

e sa

me

:m

:

:

q

n


2. Let the following vectors be defined: �⃗⃗� : ⟨−𝟑, 𝟓⟩ �⃗⃗� : ⟨𝟕, 𝟐⟩ �⃗⃗� : ⟨𝟒, −𝟗⟩

A. Simplify the following vector expression �⃗⃗� + 𝟐�⃗⃗�

and write your answer in component form.

B. Simplify the following vector expression ‖�⃗⃗� − �⃗⃗� ‖

(i.e. What is the magnitude of the resultant vector?)

C. Find the direction of the vector expression

�⃗⃗� + �⃗⃗� + �⃗⃗� .

D. Simplify the following vector expression

2�⃗⃗� + �⃗⃗� and write your answer in polar form.

3. Let the following vectors be defined: �⃗⃗� : ⟨𝟒, 𝟑𝟎°⟩ �⃗⃗� : ⟨𝟑, 𝟏𝟐𝟎°⟩

A. Rewrite vector �⃗⃗� in rectangular form.

B. Rewrite vector �⃗⃗� in rectangular form.

C. Determine �⃗⃗� + �⃗⃗� in rectangular form.

D. Determine �⃗⃗� + �⃗⃗� in polar form.

2a. 2b.

2c. 2d.

3a.

3b.

3c.

3d.


4. Let the following vectors be defined: �⃗⃗� : ⟨𝟒, 𝟑𝟎°⟩ �⃗⃗� : ⟨𝟑, 𝟏𝟐𝟎°⟩

Determine �⃗⃗� + �⃗⃗� in polar form.

(Use law of Cosines to determine your answer.)

5. Let the following vectors be defined: �⃗⃗� : ⟨𝟐, 𝟒𝟎°⟩ �⃗⃗� : ⟨𝟒, 𝟑𝟒𝟎°⟩

A. Rewrite vector �⃗⃗� in rectangular form.

B. Rewrite vector �⃗⃗� in rectangular form.

C. Determine �⃗⃗� − �⃗⃗� in rectangular form.

D. Determine �⃗⃗� − �⃗⃗� in polar form.

E. Determine 𝟓�⃗⃗� in polar form.

6. Let the following vectors be defined: �⃗⃗⃗� : ⟨𝟑, 𝟏𝟑𝟒°⟩ �⃗⃗� : ⟨𝟔, 𝟑𝟑°⟩

Determine �⃗⃗⃗� + �⃗⃗� in polar form.

(Use law of Cosines to determine your answer.)

5a.

5b.

5c.

5d.


4.

5e.

6.

7. Let the following vectors be defined: �⃗⃗� : ⟨𝟔, 𝟑𝟖°⟩ �⃗⃗� : ⟨𝟐, 𝟏𝟔𝟑°⟩

Sketch the vector expression �⃗⃗� + 𝟐�⃗⃗� in polar form. Start the vector sketch with the point shown below.

8. Sketch a vector graph of two movers trying to move a washing machine up a set of stairs. The first mover is pushing at an angle of 165° and a force of 140 pounds. The second mover is pulling at an angle of 105° and a force of 130 pounds. What is the estimated magnitude and direction of the resultant force based on your drawing?

If the movers require a minimum of 200 pounds of force at roughly a 135° angle, do you think they will have enough force to move the washing machine up the stairs?

●


9. A boy is swimming across a river. The boy’s path makes a 36° angle with the river bank and the boy is swimming slightly

upstream (against the current). The boy is swimming at a rate of 4 feet per second and the current is flowing at 2 feet

per second. How fast is the boy actually moving and in what direction?

If the river is roughly 31 feet across in width, how many seconds will it take the boy to get across the river?

10. Two tow trucks are trying to pull a car out of the mud at the same time. The first tow truck is pulling the car due East

with a force of 900 Newtons. The second truck is pulling the same car 34° North of East from the first tow truck with a

force of 1400 Newtons. Which direction will the car most likely move and how many Newtons is being applied to the car?

36°


1. Sec 7.3 – Vector Matrices and Transforms Name:

1. Let the following vectors be defined: �⃗⃗� : ⟨−𝟑, 𝟓⟩ �⃗⃗� : ⟨𝟕, 𝟐⟩ �⃗� : ⟨𝟒, −𝟗⟩ �⃗⃗� : ⟨−𝟐,−𝟏⟩

A. Rewrite each vector as a column & row matrix. Then, store each as a column matrix in your Graphing Calculator.

(Press and change the dimensions to 2 x 1 and enter the vector components)

B. Using your Graphing Calculator evaluate the following:

(i) 𝑎 − 3�⃗� (ii) −4𝑑 (iii) 2�⃗� + 3𝑎 − 4𝑐

e.g.

2. Using matrices to define vectors can be helpful to create transformations of vectors. Graph each matrix

expression as a vector in standard position and describe how each vector compares to vector 𝒗.⃗⃗⃗

A. The vector 𝑣 : [3 1

]

B. The vector �⃗� : [0 −11 0

] ∙ [3 1

]

C. The vector �⃗⃗� : [−2 00 2

] ∙ [3 1

]

MATRIX EDIT


Here are the 2-dimensional vector transformation matrices.

Reflection Matrices Reflect over y-axis Reflect over x-axis Reflect over y = x Reflect over y = – x

1 0

0 1

1 0

0 1

0 1

1 0

0 1

1 0

Rotation Matrices about the Origin Rotate by 90° Rotate by 180° Rotate by 270° Rotate by θ°

0 1

1 0

1 0

0 1

0 1

1 0

cos sin

sin cos

3. Given that vector 𝒗 ⃗⃗ ⃗ can be defined as 𝒗 ⃗⃗ ⃗: [𝒂𝒃] describe what transformations take place with each of the

following (compared to 𝒗 ⃗⃗ ⃗)

A. The vector 𝑝 : 3 ∙ [0 1

−1 0] ∙ [

𝑎 𝑏

] B. The vector 𝑞 : [4 00 −4

] ∙ [𝑎 𝑏

]

4. Given that vector 𝒖 ⃗⃗ ⃗ can be defined as 𝒖 ⃗⃗ ⃗: [𝟐𝟔] , perform the following transformations on the vector.

A. Rotate vector �⃗� 180° about the origin

and decrease the magnitude by ½ .

B. Reflect vector �⃗� over the y – axis. Then, rotate 50°

about the origin and finally, dilate it by a factor of 2.


1. Sec 7.4 – 3-Dimensional Vectors Name:

1. Determine the rectangular form

of the 3 dimensional vector 𝐴𝐵⃗⃗⃗⃗ ⃗.

Given that the points A and B are

defined as 𝐴: (2, −3,−4) and

𝐵: (4, 5, 5) as shown in the graph.

2. Find the rectangular form of the vector from point A to point B.

a. A( −3, 2, 5) B(6, −2, 2) b. A( 1, −3, 0) B(8, −3, 1) c. A( 4, −2, 7) B(4, 4, −6)

3. Sketch a graph of the 3 dimensional

vector shown below in standard position:

𝑣 : ⟨−2, 4, 2⟩


A

B

4. Describe the vector CD in standard position at the right using component

vectors , ,i j k .

5. Find the magnitude of the vector in standard position.

6. Describe the vector GH in standard position at the right using component

vectors , ,i j k .

7. Find the magnitude of the vector in standard position.

8. Using algebra find the resultant vector of CD GH from problems #4 and #6.

9. Given the vectors 2,1,5v and 3, 2, 4w find the following:

a. 2 3v w b. 3v w c. 2 4v w

2

-2

5

2

-2

y

x

z

CD

2

-2

-5

2

-2

y

x

z

GH


10. Determine the spherical form of

the 3 dimensional vector

𝑣 : ⟨−2, 2,4⟩

11. Determine the spherical form of

the 3 dimensional vector

�⃗� : ⟨−4,−3, 2⟩


12. A satellite is located in space is orbiting space about

250 miles above Maine. The monitoring station in

Houston, Texas can tell the satellite is 1450 miles due

East, 1280 miles due North, and 250 miles above the

Earth. If we simplify the problem and consider all

measures given to be linear (i.e. assuming the

curvature of the Earth wasn’t used to determine the

distances), what is the distance in miles from the

monitoring station to the satellite?

Can you determine the azimuthal

angle (i.e. horizontal angle) using

degrees East of North using the

monitoring station as the origin?

Can you determine the angle of elevation the satellite

dish should be set to in order to aim at the satellite?

13. A blimp air ship is ascending vertically at 3 miles per

hour due to its buoyancy. The ship is also being

propelled by a propeller at 25 miles per hour due East

and a wind is blowing due South at a rate of 5 miles

per hour. How fast is the air ship actually moving?

Can you determine the azimuthal

angle heading (i.e. horizontal angle)

using degrees East of North using

the blimp as the origin?

3 m

ph

25 mph E

S

UP


1. Sec 7.5 – Advanced Vector Operations Name:

The dot product of two vectors can be calculated two ways:

�⃗⃗� ∙ �⃗⃗� = ‖�⃗⃗� ‖ ‖�⃗⃗� ‖ 𝐜𝐨𝐬(𝜽)

The same value can also be determined by finding the sum of the

products of each corresponding element.

�⃗⃗� ∙ �⃗⃗� = 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 + ⋯+ 𝒖𝒏𝒗𝒏

For example, let �⃗� : ⟨−4,6⟩ and 𝑣 : ⟨3,2⟩ as shown at the right.

�⃗� ∙ 𝑣 = (−4)(3) + (6)(2) = −12 + 12 = 0

Notice that the if the dot product of two non-zero vectors is 0 then the two

vectors are perpendicular. This could be proven to some extent for a

2-dimensional vectors without too much difficulty if we can assume that perpendicular lines have negative reciprocal slopes.

Consider vector 𝑎 : ⟨𝑥, 𝑦⟩ would have a slope of 𝑦

𝑥 and a vector that is perpendicular must then have a slope

of − 𝑐𝑥

𝑐𝑦 where c is some constant, which would suggest that a perpendicular vector would be of the form

�⃗� : ⟨−𝑐𝑦, 𝑐𝑥⟩ which would have a dot product of 𝑎 ∙ �⃗� = (𝑥)(−𝑐𝑦) + (𝑦)(𝑐𝑥) = −𝑐𝑥𝑦 + 𝑐𝑥𝑦 = 0

1. Find the dot product of the following set of vectors, graph the vectors, and determine which are perpendicular.

a. 𝑝 : ⟨−4,3⟩

and

𝑞 : ⟨ 4 ,5 ⟩

b. �⃗⃗� : ⟨−8,2⟩

and

�⃗� : ⟨ 1, 4⟩

c. �⃗� : ⟨−4,−3,2⟩ and 𝑣 : ⟨−3,2, −3⟩


Magnitude

of vector �⃗�

Magnitude

of vector 𝑣 Cosine of the angle,,

between the two vectors

2. Answer the following using the dot product.

a. Consider the vectors 𝑝 : ⟨−6 ,4 ⟩ and

𝑞 : ⟨ 6 , 𝑎 ⟩. Using the dot product, what

value of ‘a’ would ensure the vectors are

perpendicular?

b. Consider the vectors �⃗⃗� : ⟨2, 3, −1 ⟩ and �⃗� : ⟨ 5, 𝑏 , 𝑐 ⟩.

Using the dot product, determine 2 sets of values that

would ensure the vectors are perpendicular. (Can you

determine a general solution to ensure the vectors

are perpendicular?)

The cross product of two 3-d vectors can be calculated two ways:

�⃗⃗� × �⃗⃗� = (‖�⃗⃗� ‖ ‖�⃗⃗� ‖ 𝐬𝐢𝐧(𝜽)) ∙ 𝒏

The same value can also be determined by finding the determinant of

the matrix created by

�⃗⃗� × �⃗⃗� = 𝒅𝒆𝒕 |𝒊 𝒋 �⃗⃗�

𝒖𝟏 𝒖𝟐 𝒖𝟑

𝒗𝟏 𝒗𝟐 𝒗𝟑

|

For example, let �⃗⃗� : ⟨𝟏, −𝟐, 𝟏⟩ and �⃗⃗� : ⟨𝟐, −𝟏, 𝟏⟩ as shown at the right.

�⃗� × 𝑣 = 𝑑𝑒𝑡 |𝑖 𝑗 �⃗�

1 −2 12 −1 1

|

= |𝑖 𝑗 �⃗�

1 −2 12 −1 1

| 𝑖 𝑗 1 −22 −1

This cross product creates a new vector that is perpendicular to both of the original vectors simultaneously which

we could verify with the dot product:

�⃗� ∙ 𝑐 = ⟨1, −2, 1⟩ ∙ ⟨−1, 1, 3⟩ = (1)(−1) + (−2)(1) + (1)(3) = −1 + −2 + 3 = 0

𝑣 ∙ 𝑐 = ⟨2, −1, 1⟩ ∙ ⟨−1, 1, 3⟩ = (2)(−1) + (−1)(1) + (1)(3) = −2 + −1 + 3 = 0

Since the dot products are each zero that suggests that each of the two sets of vectors are perpendicular but the

dot product of �⃗� ∙ 𝑣 = 5 so the original vectors �⃗� and 𝑣 are not perpendicular

Magnitude

of vector �⃗�

Magnitude

of vector 𝑣

Sine of the angle,,

between the two vectors

The unit vector at right

angles to both vectors

(−2𝑖 + 2𝑗 − 1�⃗� ) − (1𝑗 − 1𝑖 − 4�⃗� ) = −1𝑖 + 1𝑗 + 3�⃗� = ⟨−1,1,3⟩ = 𝑐


3. Find the cross product of the following set of vectors

a. �⃗⃗� : ⟨3, −4,2⟩ and �⃗� : ⟨5, 2, 1⟩

b. 𝑝 : ⟨3, −1,2⟩ and 𝑞 : ⟨−6, 2, −4⟩

4. Consider the vectors �⃗� : ⟨3,2,2⟩ and 𝑣 : ⟨4, 1, 2⟩

a. Determine �⃗� × 𝑣

b. Determine 𝑣 × �⃗�

c. Is the cross product commutative?

The angle between two vectors can be calculated using the Law of Cosines:

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃 𝒄𝒐𝒔(𝑪)


The formula can be extended to higher dimensions as well. Using the formula complete the problems below.

𝒄𝒐𝒔(𝝑) = �⃗⃗� ∙ �⃗⃗�

‖�⃗⃗� ‖ ‖�⃗⃗� ‖

5. Graph the following vectors and find the angle between the following set of vectors

a. 𝑎 : ⟨4, −2⟩ and �⃗� : ⟨−1,4⟩

b. �⃗⃗� : ⟨−6,−4⟩ and �⃗� : ⟨−2,7⟩

c. �⃗� : ⟨2, −2, 3⟩ and 𝑣 : ⟨−3, 2, 1⟩


1. Sec 7.6 – Complex Numbers (Review) Name:

For possibly several well over a thousand years, mathematicians disregarded the idea of attempting to find the square root of a negative number because at the time they thought it was absurd and useless. There is no “real”

number that when multiplied by itself is a negative value which makes finding something like √−4 seemingly impossible.

2 ∙ 2 = 4 −2 ∙ −2 = 4 0 ∙ 0 = 0 −1 ∙ −1 = 1

According to some references the first written suggestion of attempting to find the square root of a negative number may have dated all the way back to 50 A.D when Heron of Alexandria was trying to determine the volume of an impossible section of the pyramid. The first substantial noted works about finding the square root of a negative number didn’t appear again until the 1500’s when there was a math duel to see who could solve general cubic equations more effectively between three Italian mathematicians, Cardano and Ferrari versus Tartaglia. Ferrari was a student of Cardono and stood in for him during the duel. Cardano eventually published these findings in the Ars Magna. There was some argument as to who was first and who was better but the end result was that the solutions required taking square roots of negative numbers. Later in the 1600’s it was Rene Descartes, considered the father of analytical geometry, that

accidentally coined the term ‘imaginary’ to represent the number √−1 as well as the

standard form for complex numbers of 𝑎 + 𝑏√−1 . Finally, it was Leonhard Euler that gave us the standard notation of i to represent the

number √−1 and finalize the complex notation we use today of 𝑎 + 𝑏𝑖 . Today, the understanding of imaginary numbers are commonly used in several engineering studies of such topics as force stresses, electrical engineering, and resonance.

1. Find all square roots of each of the following and circle the principal square root.

a. 196 b. 1 c. 0 d. – 1 e. – 25

2. What does i represent? Which mathematician was the first to call the number imaginary?

What is standard form? Which mathematician was the first to use the symbol i?

What do you think i2 should be? Can you explain why it MUST be defined?

3. Simplify: a. i3 = b. i4 = c. i11 =

d. i100 = e. i103 = f. i131 =


4. Simplify the following:

a. 45 b. 722

5. Add/Sub and simplifying the following and write the answer in standard form (a + bi):

a. 2 5i i b. ii 6253 c. ii 2832

6. Add/Sub and simplifying the following and write the answer in standard form (a + bi):

a. 25653 i b. 3241218

7. Mult/Div and simplify the following:

a. 2 5i i c. 123 d. 15 20i i

e. ii 8742 f. 243 i g. ii 5353


8. What are the complex conjugates of each of the following?

a. i64 b. i8 c. 6


b.

10

6

i

i b.

2 5

3

i

i

c.

6 5

4

i

i

d.

3 2

5 3

i

i



a.

4 2

2 3

i

i

b.

6 2 3 3

3

i i

i

11. Solve the following complex equations

a. 8 12 4 2i x i y b. 4 3 2 2 2i i x i y

c. 18 3 9i x y i


1. Sec 7.7 – Advanced Vector Operations Name:

The complex plane is a graphical interpretation of the complex

numbers. The x-axis becomes the “REAL NUMBER” axis and the

y-axis becomes the “IMAGINARY NUMBER” axis. Using this

concept, we can plot complex numbers using a rectangular

Cartesian coordinate system.

Consider the complex numbers

𝒂 = 𝟔 + 𝟐𝒊

𝒃 = −𝟐 − 𝟓𝒊

We can find their sum by using a vector style approach.

𝒂 + 𝒃 = (𝟔 + −𝟐) + (𝟐 + −𝟓)𝒊 = 𝟒 − 𝟑𝒊

We can also use a graphical head to tail vector style approach:

The resultant vector is located at 𝟒 − 𝟑𝒊

1. Plot the following complex numbers or show the suggested expression.

A. 4 − 3𝑖

B. −4𝑖

C. 𝑐 = −3 + 2𝑖 & 𝑑 = 5 + 2𝑖

𝑐 + 𝑑


2. Rewrite each of the following complex numbers in polar form and graph. A. Rewrite the complex number 4 + 3𝑖 in polar (cis) form and re-graph it on the polar graph paper.

B. Rewrite the complex number −4 + 2𝑖 in polar (cis) form and re-graph it on the polar graph paper.

3. Rewrite each of the following complex numbers in rectangular form and graph. A. Rewrite the complex number 4(cos 250° + 𝑖 sin 250°) in rectangular form and re-graph it.

B. Rewrite the complex number 3 𝑐𝑖𝑠 320° in rectangular form and re-graph it


4. Determine the following. A. Evaluate |−7 + 4𝑖|

B. Find the argument of 5 − 3𝑖

C. Find the modulus of 12 − 5𝑖

D. Find the exact rectangular form of 4 𝑐𝑖𝑠 120°

5. Consider the complex numbers 𝒂 = 𝟑 𝐜𝐢𝐬 𝟔𝟎° and 𝒃 = 𝟐 𝐜𝐢𝐬 𝟏𝟓𝟎°

A. Graph each complex number.

B. Rewrite each complex number in

approximate rectangular form.

C. Algebraically, multiply the two complex numbers in rectangular form.

D. Rewrite the product ab in polar form.

E. Graph the product of ab.


6. Using this Evaluate the following and leave your answer in polar form check your answers with your calculator.

A. Evaluate (4 𝑐𝑖𝑠 100°)(3 𝑐𝑖𝑠 220°)

B. Evaluate (√2 𝑐𝑖𝑠 230°)(√18 𝑐𝑖𝑠 200°)

C. Evaluate (7 𝑐𝑖𝑠 130°)2

D. Evaluate (20 𝑐𝑖𝑠 260°)

(5 𝑐𝑖𝑠 150°)

E. Evaluate √25 𝑐𝑖𝑠 250°

F. Evaluate √8 𝑐𝑖𝑠 360°3

G. Evaluate (8 𝑐𝑖𝑠 70°) + (2 𝑐𝑖𝑠 210°)


a. describe vector...1. sec 7.1 – basic vector forms name: 1. describe each of the following...

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