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PreCalculus NotesMAT 129

Chapter 10: Polar Coordinates; Vectors

David J. Gisch

Department of MathematicsDes Moines Area Community College

October 25, 2011

1 Chapter 10Section 10.1: Polar CoordinatesSection 10.3: The Complex Plane; De Moivre’s TheoremSection 10.4: VectorsSection 10.5: The Dot Product

Section 10.1: Polar Coordinates

Summary

We will learn about Polar coordinates; convert rectangularcoordinates to polar coordinates and vice versa; and transformequations from polar to rectangular form.

Radians vs. Degrees

Polar Coordinates

Figure: Graph of polar coordinates (2, π/4)

Example

Graph the polar coordinates(4,π

2

),

(− 2,

5π

4

),

(3,−2π

3

),

Several Forms of a Single Point

Conversion from polar Coordinates to Rectangular

Transform Polar Equations to Rectangular Form

Identities

r2 = x2 + y2

x = r cos θ

y = r sin θ

Example

Change r = 4 cos θ to rectangular form.

Example

Change r = 41−cos θ to rectangular form.

Example

Change y = 3x + 2 to polar form.

Example

Change x2 + y2 = 1 to polar form.

Section 10.3: The Complex Plane; De Moivre’sTheorem

Summary

We will plot polar coordinates in the complex plane; findproducts and quotients of complex numbers; and use DeMoirve’s theorem to find roots of complex numbers.

Definition

We can write a complex number a + bi as z = x + yi . The magnitude ormodulus of z , denoted as |z |, is the distance of z from the origin

|z | =√

x2 + y2

In other words, |z | = r .

Note

Recall that any complex number a + bi has the conjugate a− bi . Thus,any complex number z = x + yi has the conjugate z̄ = x − yi .

Definition

If r ≥ 0 and 0 ≤ θ ≤ 2π, the complex number z = x + yi may be writtenin polar form as

z = x + yi = r cos θ + (r sin θ)i = r(cos θ − i sin θ)

Note

If x = r cos θ then cos θ = x/r and likewise sin θ = y/r .

Example

Write z =√

2− i in polar form.

Example

Write z = 2 +√

3i in polar form.

Products and Quotients

Theorem

Let z1 = r1(cos θ1 − i sin θ1) and z2 = r2(cos θ2 − i sin θ2)

z1z2 = r1r2[

cos(θ1 + θ2)− i sin(θ1 + θ2)]

z1z2

=r1r2

[cos(θ1 − θ2)− i sin(θ1 − θ2)

]

Example

Multiply and divide z1 = 3(cos 30◦ − i sin 30◦) andz2 = 2(cos 100◦ − i sin 100◦) in polar form.

Example

Multiply and divide z1 =√

3 + 2i and z2 = −2√

2 + 2√

2i in polar form.

De Moivre’s Theorem

Theorem

Let z = r(cos θ − i sin θ), then

zn = rn[

cos(nθ)− i sin(nθ)]

Example

If z = 2(cos 30◦ − i sin 30◦), calculate z6.

Example

Calculate (1 + i)5 and write your answer in a + bi form.

Complex Roots Theorem

Let w be a complex number and n ≥ 2. Then the complex numbers thatare solutions to the equation

zn = w

are called complex nth roots of w . The complex roots can be calculated as

Theorem

Let w = r(cos θ − i sin θ), then

zk = n√

r

[cos(θ

n+

2kπ

n

)− i sin

(θn

+2kπ

n

)]where k = 0, 1, 2, . . . , n − 1.

Example

Find the complex cube roots of w = 1−√

3i .

Section 10.4: Vectors

Summary

We will plot vectors; convert vectors to various forms; andperform arithmetic of vectors.

Line, Segments, Vectors

Note

Any two vectors are considered to be equal if they have the samemagnitude and direction. There exact starting and stopping points neednot be the same. For example, all three vectors below are equal.

Adding Vectors

When you add two vectors you can think of visually as starting the secondvector from where the first one left off. We call the black vector ~v + ~w theresultant vector.

Note: Order does not matter either.

Negative Vectors

Given a vector ~v we can also consider −~v . The vector −~v has the samemagnitude as ~v but has a direction opposite of ~v .

Adding and Subtracting Vectors (Visually)

Figure: ~u + ~v

Adding and Subtracting Vectors (Visually)

Figure: ~u − ~v

Adding and Subtracting Vectors (Visually)

Figure: 2~u + ~v − ~w

Scalar Multiples

Properties of Scalar Multiples

Magnitude

Definition

The length of a vector is defined as the magnitude, written as ‖~v‖.

Note

If ‖~v‖ = 1, then we call ~v a unit vector.

Algebraic Vector

Definition

We write a vector in algebraic form as ~v = 〈x , y〉, where x and y are calledthe vector’s components. Thus, we also refer to this form as componentform.

Algebraic Vector

Theorem

If a vector ~v starts with the initial point P1 = (x1, y1), not necessarily theorigin, and terminal point P2 = (x2, y2), then ~v can be written in algebraicfrom as

~v = 〈x2 − x1, y2 − y1〉

I often think of this formula as taking “the end point minus the startpoint.”

Example

Given the following points find the vectors

P1 = (2, 3) P2 = (4,−3) P3 = (−1, 5)

1−−−→P1P2

2−−−→P3P2

3−−−→P1P3

Another Form!?

Definition

We also write vectors in algebraic form as ~v = 〈a, b〉 = ai = bj . Here isymbolizes the x-component and j symbolizes the y -component. [Notethat i is not the imaginary i here.]

Adding and Subtracting Vectors (Algebraically)

Properties

If we have two vectors ~u = ai + bj = 〈a, b〉 and ~v = ci + dj = 〈c , d〉, then

~u + ~v = (a + c)i + (b + d)j = 〈a + c , b + d〉~u − ~v = (a− c)i + (b − d)j = 〈a− c , b − d〉α~u = (αa)i + (αb)j = 〈αa, αb〉‖~u‖ =

√a2 + b2

Example

Given the following vectors calculate the sum or difference.

~u = 2i + 3j ~v = −4i + 2j

1 ~u + ~v

2 3~u − ~v

3 ‖~v‖

Unit Vector

Theorem

For any vector ~v, the vector

~u =~v

‖~v‖is a unit vector in the same direction as ~v. Note that this implies

~v = ‖~v‖~u

Example

Find a unit vector in the same direction as ~v = 4i − 3j .

Example

Show that ~v = ‖~v‖~u.

Magnitude and Direction

Vectors, when written as 〈x , y〉 can be thought of as being in rectangularform, though they do not necessarily start at the origin. Vectors can alsobe described by their magnitude, given by ‖~u‖, and direction θ, measuredfrom the positive horizontal. Thus we can treat vectors as polarcoordinates and vice versa, making the arithmetic similar.

Applications

Example

A ball is thrown with an initial speed of 25 miles per hour in a directionthat makes an angle of 30◦ with the positive horizontal. Express thevelocity vector ~v in terms of i and j. What is the initial speed in thehorizontal direction? What is the initial speed in the vertical direction?

Applications

Example

A box of supplies that weighs 1200 pounds is suspended by two cablesattached to the ceiling as shown. What is the tension on the two cables?

Applications

Example

At a picnic there is a contest, in which hoses are used to shoot water at abeach ball in three different directions. As a result there are three forcesacting upon the beach ball. The pressure and position of the hoses can begiven as

F1 = 50N at 120◦

F2 = 90N at 240◦

F3 = 70N at 0◦

Here N is the unit of force Newtons. What is the resultant vector (i.e.where is the ball going to go)?

Section 10.5: The Dot Product

Summary

We will find the dot product of two vectors; the angle betweenvectors; and determine whether vectors are parallel or orthogonal.

Dot Product

Formula

If we have two vectors ~u = ai + bj = 〈a, b〉 and ~v = ci + dj = 〈c , d〉, thenthe dot product of the two vectors is given as

~u · ~v = ac + bd

Example

Given the following vectors calculate given.

~u = 2i + 3j ~v = −4i + 2j

1 ~u · ~v

2 3~u · ~v

3 ‖~u‖

Properties of the Dot Product

Properties

If we have two or more vectors, then

~u · ~v = ~v · ~u (1)

~u · (~v + ~w) = ~u · ~v + ~u · ~w (2)

~v · ~v = ‖~v‖2 (3)

0 · ~u = 0 (4)

Law of Cosines

‖~u − ~v‖2 = ‖~u‖2 + ‖~v‖2 − 2‖~u‖‖~v‖ cos θ

Angle Between Vectors

Theorem

For any two vector ~u and ~v, the angle θ between them can be determinedby

cos θ =~u · ~v‖~u‖‖~v‖

Note that this gives the angle so that 0 ≤ θ ≤ π.

Example

Given the following vectors calculate the angle between them using the dotproduct.

~u = 4i − 3j ~v = 2i + 5j

Example

Parallel and Orthogonal Vectors

Theorem

If two vectors were parallel they would either be going in the same directionor in opposite directions, resulting in θ = 0 or θ = 180◦, respectively. Ascos(0) = 1 and cos(180) = −1 we can say that two vectors are parallel if

cos θ =~u · ~v‖~u‖‖~v‖

= ±1

Likewise, two vectors are said to be orthogonal if they form a right angle.As cos(90) = 0, we know two vector are orthogonal if

cos θ =~u · ~v‖~u‖‖~v‖

= 0

which happens if~u · ~v = 0

Example

State whether the vectors are parallel, orthogonal, or neither.

~u = 3i + 8j ~v = −2i − 16

3j ~w = −6i + 12j

1 ~u and ~v

2 3~u and ~w

Projection

Theorem

Given two vectors ~v and ~w, the projection of ~v onto ~w is

~v1 =~v · ~w‖~w‖2

~w =~v · ~w~w · ~w

~w

where ~v1 is parallel to ~w. We also have

~v2 = v − v1

which is perpendicular to ~w.

Example

Given the vectors

~u = 4i − 3j ~v = −2i + 10j

Find the projection of ~u onto ~v , ~u1, and the orthogonal vector ~u2.

In elementary physics, the work W done by a constant force F in movingan object from point A to point B is defined as

W = (magnitude of force)(distance) = ‖F‖‖−→AB‖ = F ·

−→AB

Example

The figure below shows a girl pulling a wagon with a force of 50 pounds.How much work is done in moving the wagon 100 feet if the handle makesan angle of 30◦ with the ground?