Unit 7: Cartesian Vectors

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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

Volume of a Parallelepiped

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

3 Ways to Multiply Vectors

Cross Product

Properties of the Cross Product:

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

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

u × v = -(v × u)

(u+v) × w = u × w + v × w

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

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

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

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.

Formula: a × b = |a||b|sinθ n

n is the unit vector perpendicular to both a and b

the result of the cross product is a vector

Determining the Angle Between 2 Vectors: Formula: θ = sin⁻¹(|a×b|/|a||b|)

Properties of the Dot Product:

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

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

For any vector u: u ⋅ u = |u|²

Commutative Property: u ⋅ v = v ⋅ u

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

Cartesian Vectors of the Dot Product: in R² --> a ⋅ b = a₁b₁ + a₂b₂ in R³ --> a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃

Different Types of Cases:

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

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 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 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

Formula: a ⋅ b = |a||b|cosθ

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

the result of the dot product is a scalar

the angle is between the 2 vectors

Scalar Multiplication

scalar multiplication involves multiplying a vector by a scalar

when a vector is multiplied by a scalar value = ka

Applications

Dot Product

Determining the Angle Between 2 Vectors:

Formula: cosθ = u⋅v/|u||v|

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

Scalar projection tells us about the magnitude of the projection

Scalar Projection: |projᵤv| = |v|cosθ

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

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

Formula: W = F⋅d W= |F||d|cosθ

Measured in Joules, where 1 J = 1 N⋅m

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

θ is the angle between the force and the lever arm

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

F is the applied force measured in Newtons (N)

Algebraic Vectors: 3D

Vectors in R³

Finding magnitude in 3-space: | u | = √a² + b² + c²

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)

Assigned coordinates (a,b,c)∈R³

How to go from O to P?

3) we go c units in the z-direction

2) we go b units in the y-direction

1) we go a units in the x-direction

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:

2) the position vector of point P

1) the coordinates of some point P in space

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)

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

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

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

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

2) the position vector of the point P

1) the coordinates of some point P in the plane

Algebraic Vectors: 2D

Algebraic Vectors

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

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

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