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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?