Chapter 10: Polar Coordinates and Vectors

Based on Chapter 10: Polar Coordinates and Vectors from the 8th edition of Algebra & Trigonometry, Enhanced with Graphing Utilities by Michael Sullivan and Michael Sullivan III. Table of Contents: Section 10.1: cards 1 - 10; Section 10.2: cards 11 - 24; Section 10.3: cards 25 - 39; Section 10.4: cards 40 - 71; Section 10.5: cards 72 - 89

Characteristics of rectangular coordinates

Points look like (x, y)

Plotted on Rectangular Coordinate System

Characteristics of polar coordinates

Points look like (r, θ), where θ is the angle measure and r is the radius

Plotted on the Polar Coordinate System

Can be represented in many ways

The coordinates of the pole are (0, θ)

Steps to plot the point (0, θ)

1. Draw the angle measure θ. Vertext is at the pole and initial side is at the polar axis

2. Go out a distance of r along the terminal side of θ to reach the point P

   1. If r>0, travel along the terminal side of θ

   2. If r<0, travel in the direction opposite the terminal side of θ.

Representing polar coordinates when r>0

(r, θ+2πk)

Representing polar coordinates when r<0

(-r, θ+π+2πk)

Converting polar coordinates into y

y=rsin(θ)

Converting polar coordinates into x

x=rcos(θ)

Finding the r value from rectangular coordinates

r²=x²+y²

Finding the angle measure θ from rectangular coordinates

θ=tan⁻¹(y/x)

Graphing polar equations

1. Convert to a rectangular equation

2. Square both sides or multiply both sides by r

Polar equations

An equation whose variables are polar coordinates

The graph of a polar equation consists of all points whose polar coordinates satisfy the equation

Characteristcs of r=2asin(θ)

Is a circle; radius a, center at (0, a) in rectangular coordinates

Characteristcs of r=-2asin(θ)

Is a circle; radius -a, center at (0, -a) in rectangular coordinates

Characteristics of r=2acos(θ)

Is a circle; radius a, center at (a, 0) in rectangular coordinates

Characteristics of r=-2acos(θ)

Is a circle; radius a, center (-a, 0) in rectangular coordinates

Symmetry about the x-axis

Symmetry about the y-axis

Symmetry about the origin

Graphing with a calculator

1. Solve the equation for r in terms of θ

2. Select the viewing windows in POLAR mode

3. Adjust Xmin, Xmax, Xscl and Ymin, Ymax, Yscl

4. Adjust θmin, θmax, θstep with a radian measure

Cordiod graph characteristics

r=a(1+cosθ), r=a(1-cosθ), r=a(1+sinθ), r=a(1-sinθ), where a>0 and the graph passes through the pole

Limacons without inner loop graph characteristics

r=a+bcos(θ), r=a-bcos(θ), r=a+bsin(θ), r=a-bsin(θ) where a>0, b>0, and a>b and the graph does not pass through the pole

Limacons with inner loop

r=a+bcos(θ), r=a-bcos(θ), r=a+bsin(θ), r=a-bsin(θ) where a>0, b>0, and a<b and the graph passes through the pole twice

Rose graph characteristics

r=acos(nθ), r=asin(nθ) where a≠0. If n≠0 is even, the rose has 2n petals. If n≠±1 is odd, the rose has n petals

Lemniscates graph characteristics

r²=a²sin(2θ), r²=a²cos(2θ) where a≠0 and graphs are propeller shaped

Complex number in rectangular form

z = a + bi

Complex number on a xy-plane

(a, b)

The complex plane

Horizontal axis - real axis

Vertical axis - imaginary axis

Magnitude of a complex number

The distance form the origin to the point (a, b)

Magnitude of a complex number formula

|z| = √(a² + b²)

Conjugate of a complex number

z = a + bi, then, z* = a - bi

Polar form of a complex number

z = r × (cosθ + isinθ), where r ≥ 0 and 0 ≤ θ ≤ 2π

Euler’s formula

1. e^{iθ} = cosθ + isinθ

2. r × (cosθ + isinθ) = r × e^{iθ}

3. z = r × e^{iθ}

Product of complex numbers in polar form

z₁ × z₂ = r₁ × r₂ × [cos(θ₁ + θ₂) + isin(θ₁ + θ₂)]

Quotient of complex numbers in polar form

(z₁ ÷ z₂) = (r₁ ÷ r₂) × [cos(θ₁ - θ₂) + isin(θ₁ - θ₂)]

Product of complex numbers in exponential form

z₁ × z₂ = (r₁ × r₂) × e^{i × (θ₁ + θ₂)}

Product of complex numbers in exponential form

(z₁ ÷ z₂) = (r₁ ÷ r₂) × e^{i × (θ₁ - θ₂)}

Exponential DeMoivre’s theorem

zⁿ = rⁿ × e^{i × (nθ)}

Polar DeMoivre’s Theorem

zⁿ = rⁿ × [cos(nθ) + isin(nθ)]

nth root formula

ⁿ√{z} = ⁿ√{r} × (cos({θ ÷ n} + {360k ÷ n}) + isin({θ ÷ n} + {360k ÷ n})) where k = 0, 1, 2, … n - 1

Characteristics of a vector

1. A directed line segment that has both direction and magnitude

2. Magnitude: the length

3. Direction: direction it is pointing/heading

Notations for vectors

1. v - a bold lower-case letter

2. →v - a lower-case letter with an arrow pointing to the right

3. ^v - a lower-case letter with an arrow pointing up

Triangle method for adding vectors

Begins at the initial side of u and stops at the terminal side of v

Parallelogram method for adding vectors

Begins at the shared initial side of u and v and runs along the diagonal of the parallelogram

Resultant vector

The sum of vectors, u + v

Additive properties of vectors

1. Commutative: v + w = w + v

2. Associative: u + (v + w) = (u + v) + w

The zero vector property

v + 0 = 0 + v = v

Zero vector characteristics

The magnitude is 0 and no direction

Negative vectors characteristics

If v is a vector, then -v is a vector with the same magnitude but opposite direction

Adding negative vectors

v + (-v) = 0

Subtracting negative vectors

v - w = v + (-w)

Scalar multiple definition

If α is a scalar (|R) and v is a vector, then αv is a scalar multiple of v

Scalar multiplication

1. If α > 0, the product αv is the vector whose magnitude is α times the magnitude of v and whose direction is the same as v

2. If α < 0, the product αv is the vector whose magnitude is |α| times the magnitude of v and whose direction is opposite of v

3. If α = 0, or v = 0, then αv = 0

Magnitude of vectors

Length of a vector, should be zero or positive

Notation of magnitude

‖v‖ or |v|

Characteristics of magnitudes

1. |v| ≥ 0

2. |-v| = |v|

3. |v| = 0 if v = 0

4. |αv| = |α| × |v|

Unit vector

Has a magnitude of 1 unit and is denoted by →u or u

Position vector definition

Vectors that start at the origin

Position vector notation

v = <a, b>

Position vector characteristics

1. a and b are real numbers

2. a and b are components of the vector

3. The vector must start at the origin. a describes the horizontal movement and b describes the vertical movement

4. When v is represented as a position vector v = <a, b>, then the terminal point of v will be at the ordered pair (a, b)

Position vector theorem

Suppose v has an initial point P₁ = (x₁, y₁), not necessarily at the origin, and terminal point P₂ = (x₂, y₂). If v = P₁⁻P₂⁻, then v is equal to the position vector: v = <x₂ - x₁, y₂ - y₁>

Equality of vectors definition

Horizontal and vertical components are equal

Equality of vectors theorem

If v = <a₁, b₁> and w = <a₂, b₂>, then v = w if a₁ = a₂ and b₁ = b₂

Position vector in component form

v = ai + bj

Meaning of vector i

Horizontal movement of v

Meaning of vector j

Vertical movement of v

Adding vectors algebraically

If v = a₁i + b₁j = <a₁, b₁> and w = a₂i + b₂j = <a₂, b₂>, α ∈ scalar, then:

1. v + w = (a₁ + a₂)i + (b₁ + b₂)j

2. v + w = <a₁ + a₂, b₁ + b₂>

Subtracting vectors algebraically

If v = a₁i + b₁j = <a₁, b₁> and w = a₂i + b₂j = <a₂, b₂>, α ∈ scalar, then:

1. v - w = (a₁ - a₂)i + (b₁ - b₂)j

2. v - w = <a₁ - a₂, b₁ - b₂>

Multiplying vectors algebraically

If v = a₁i + b₁j = <a₁, b₁> and w = a₂i + b₂j = <a₂, b₂>, α ∈ scalar, then:

1. αv = (αa₁)i + (αb₁)j

2. αv = <αa₁, αb₁>

Magnitude of a vector

|v| = √{a² + b²}

Unit vector in the same direction as vector v

1. u = {v ÷ |v|}

2. v = |v| × u

3. u = <{a ÷ |v|}, {b ÷ |v|}>

Vector from magnitude and direction

v = |v| × (cos(α)i + sin(α)j)

Multiplying vectors using Scalar Product

Formula: αv

Output: vector

Multiplying vectors using Dot Product

Formula: v ⋅ w

Output: scalar

Multiplying vectors using Cross Product

Formula: v × w

Output: vector

Dot Product Theorem

v ⋅ w = (a₁ * a₂) + (b₁ * b₂)

Commutative property of Dot Product

u ⋅ v = v ⋅ u

Distributive property of Dot Product

u ⋅ (v + w) = u ⋅ v + u ⋅ w

Square property of Dot Product

v ⋅ v = a² + b²

Zero property of Dot Product

0 ⋅ v = 0

3D Vector

1. Third component is k

2. v = ai + bj + ck

Dot Product Theorem with three components

v ⋅ w = (a₁ * a₂) + (b₁ * b₂) + (c₁ * c₂)

Property of dotting a vector by itself

v ⋅ v = |v|²

Angle between vectors

1. An angle between any two vectors that share an initial point

2. cos(θ) = {u ⋅ v} / {|u| * |v|}

Parallel vectors

1. Vectors that are moving in the same or opposite direction

2. The angle is θ = 0° or θ = 180°

3. v = αw, α ≠ 0

Orthogonal vectors

1. Vectors that are perpendicular to each other

2. The angle is θ = 90°

3. v ⋅ w = 0

Work in the same direction

1. Work done by a constant force F moving an object from A to B, when the force is applied in the same direction as the objecting traveling

2. Work = (magnitude of force) (distance traveled)

3. W = |F| * |d|

Work in different directions

1. Work done by a constant force F moving an object from A to B, when the force is not in the same direction

2. Work = (magnitude of force) (distance traveled) (cosine of angle)

3. W = |F| * |d| * cos(θ)

4. W = F ⋅ d

5. W = |proj{d} F| * |d|

Projection vector

Projection of v onto w is: proj{w} v = {v ⋅ w} / |w|² * w

Decomposition of vectors

1. v₁ = proj{w} v = {v ⋅ w} / |w|² * w

2. v₂ = v - v₁