Unit 7:
Cartesian Vectors

Algebraic Vectors: 2D

Geometric Vectors:

Vectors with no fixed location

Is written as a magnitude + a direction

e.g., a --> |a| = 2km [N26°E]

Algebraic Vectors

Vectors drawn on a coordinate plane,
the tail is at the origin (0,0)

Can be written using polar coordinates:

angle is measured from the terminal arm
(positive x-axis)

Can be written using Cartesian coordinates:
- in coordinates form
- in unit vectors form (also known as component form)

coordinates form:
e.g., a = (3, 5)

unit vector form:
e.g., a = 3î + 5ĵ

î = unit vector of x-axis (1,0)
ĵ = unit vector of y-axis (0,1)

î and ĵ are also known as
the vector components

To calculate the magnitude,
use the Pythagorean Theorem:
a² + b² = c²

Algebraic Vectors: 3D

Vectors in R²

The coordinates are a
pair of real numbers (a,b)

u = any vector in the plane, but position
it so that its tail is at the origin and its
head is at some point P(a,b)

u can also be interpreted as
the position vector, written
as u = OP = (a,b)

(a, b) are known as the
scalar components
and has 2 meanings:

1) the coordinates of some
point P in the plane

2) the position vector
of the point P

P (position vector) starts at the
origin, and ends at any other
random point

let u = r
the coordinates of P (a,b)

Vectors in R³

Points in 3D space can be described using
ordered triples of real numbers

u = any vector in 3-space but position
it so that its tail is at the origin and its
head is at some point P(a,b,c)

(a, b, c) has 2 meanings:

1) the coordinates of some
point P in space

2) the position vector of
point P

Assigned coordinates (a,b,c)∈R³

How to go from O to P?

1) we go a units in the
x-direction

2) we go b units in the
y-direction

3) we go c units in the
z-direction

The unit vectors in 3-space are called the
standard basis vectors: î, ĵ, and k̂

Written as:
î = (1, 0, 0)

ĵ = (0, 1, 0)

k̂ = (0, 0, 1)

Finding magnitude in 3-space:

| u | = √a² + b² + c²

Applications

Cross Product:

Torque:
Think of it as a physical quantity that describes the
rotational of turning effort of a force

Formula:
τ = |r × F|
τ = |r||F|sinθ

F is the applied force
measured in Newtons (N)

r is the vector determined by the
lever arm from the axis of rotation
measured in metres (m)

θ is the angle between the
force and the lever arm

Dot Product

Work:
The product of the magnitude of the displacement travelled by an object, and the magnitude of the force applied in the direction of the motion

Measured in Joules, where 1 J = 1 N⋅m

Formula:
W = F⋅d
W= |F||d|cosθ

Projection:
Think of projection like a shadow, where the projection of v onto u is the shadow that v casts on u

Vector Projection:
|projᵤv| = (v⋅u/|u||u|)u

|projᵥu| = (u⋅v/|v||v|)v

Scalar projection tells us
about the magnitude of
the projection

Scalar Projection:
|projᵤv| = |v|cosθ

Determining the Angle Between 2 Vectors:

Formula:
cosθ = u⋅v/|u||v|

3 Ways to Multiply Vectors

Scalar Multiplication

when a vector is multiplied by
a scalar value

= ka

scalar multiplication involves multiplying
a vector by a scalar

Dot Product

Formula:

a ⋅ b = |a||b|cosθ

the angle is between the 2 vectors

the result of the dot product
is a scalar

a and b are two non-zero vectors
arranged tail-to-tail forming an angle

Different Types of Cases:

Case 1: θ is 0 degrees

e.g., |a| = 10, |b| = 4, θ = 0 degrees

a ⋅ b = |a||b|cosθ
a ⋅ b = 10(4)cos0
a ⋅ b = 10(4)(1)
a ⋅ b = 40 units

cos0 on the unit circle = 1

Case 2: θ is acute

e.g., |a| = 10, |b| = 4, θ = 60 degrees

a ⋅ b = |a||b|cosθ
a ⋅ b = 10(4)cos60
a ⋅ b = 10(4)(1/2)
a ⋅ b = 20 units

cos60 on the unit circle = 1/2

As a result, the dot product between
a and b will always greater
than zero (positive) if the angle is
less than 90 degrees

Case 3: θ is 90 degrees

e.g., |a| = 10, |b| = 4, θ = 90 degrees

a ⋅ b = |a||b|cosθ
a ⋅ b = 10(4)cos90
a ⋅ b = 10(4)(0)
a ⋅ b = 0 units

cos90 on the unit circle = 0

Under this case, the vectors are
perpendicular/orthogonal.

As a result, the dot product between
aand b will always be zero if the
angle is 90 degrees

Case 4: θ is obtuse

e.g., |a| = 10, |b| = 4, θ = 150 degrees

a ⋅ b = |a||b|cosθ
a ⋅ b = 10(4)cos150
a ⋅ b = -34.64 units

cos90 on the unit circle = -√3/2

As a result, the dot product between
a and b will always less than
zero (negative) if the angle is
greater than 90 degrees

Cartesian Vectors of the Dot Product:

in R² --> a ⋅ b = a₁b₁ + a₂b₂
in R³ --> a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃

Properties of the Dot Product:

u ⋅ v = 0 when the vectors
are perpendicular to each other

Commutative Property:
u ⋅ v = v ⋅ u

For any vector u:
u ⋅ u = |u|²

Associative Property:
(ku) ⋅ v = k(u ⋅ v) = u ⋅(kv)

Distributive Property:
u(v ⋅ w) = u ⋅ v + u ⋅ w

Cross Product

Formula:

a × b = |a||b|sinθ n

the angle is between the 2 vectors

Determining the Angle Between 2 Vectors:

Formula:
θ = sin⁻¹(|a×b|/|a||b|)

the result of the cross product
is a vector

n is the unit vector perpendicular to both a and b

A vector called the "cross product", has a magnitude equal to the area of the parallelogram the 2 vectors create, and is perpendicular to the 2 vectors.

The Right Hand Rule:
it's used to determine the direction of the
cross product vector

1) index finger is the direction of the 1st vector (a)
2) middle finger is the direction of the 2nd vector (b)
3) the direction where the thumb points gives the
direction of a×b

Cartesian Vectors of the Cross Product:

a × b = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]
a × b = (a₂b₃ - a₃b₂)î + (a₃b₁ - a₁b₃)ĵ + (a₁b₂ - a₂b₁)k̂

Properties of the Cross Product:

if u and v are non-zero, u×v 0
only if there's a scalar such as
u = mv

(u+v) × w = u × w + v × w

u × v = -(v × u)

Associative Property:
k(u×v) = (ku)×v = u×(kv)

Distributive Property:
u×(v + w) = u × v + u × w

Volume of a Parallelepiped

Formula:
V = A(base) ⋅ height
V = |a ⋅ (b×c)|

Number of axes differ between 2D space, and 3D space:

- in 2D space, there are 2 axes: the x-axis, and the y-axis
- in 3D space, there are 3 axes: the x-axis, the y-axis, and the z-axis

Unit 7: Cartesian Vectors