Unit 3 Dot Product / Cross Product

Question 1

What is Dot Product used for?

Answer 1

Dot Product is used to show us how the amount of magnitude a particular force has in several arbitrary directions.

Dot Product can also be used to find the angle between two vectors given either their Unit or Force Vector has already been defined.

Question 1

What is the formula for Dot Product ?

Answer 1

F₁⋅F₂
= (F₁x )(F₂x) + (F₁y )(F₂y ) + (F₁z )(F₂z)

Question 2

How do you find the magnitude of a force parallel to an arbitrary direction ?

Answer 2

By dotting the force by the Unit Vector of the arbitrary direction:

F₁⋅û₂ = F∥

Question 3

How do you find the Angle between two vectors ?

Answer 3

By manipulating the Dot Product formula to solve for θ:

F₁⋅F₂ = |F₁||F₂|cos(θ)

cos⁻¹[(F₁⋅F₂) / (|F₁||F₂|)] = θ

Question 4

How do you find the magnitude of a force perpedicular to an arbitrary direction.

Answer 4

Start by using dot product of the force dotted to the arbitrary direction:

F₁⋅û₂ = F∥

Using F₁ as the hypotneuse, create a line perpedictular to our new vector F∥.

The angle between F₁ and F∥ (θ) will be found by using our manipulated Dot Product formula with F₁ and û₂ as our variables:

cos⁻¹[(F₁⋅û₂) / (|F₁||û₂|)] = θ

Using our new triangle along with its angle, we can now solve for the perpendicular vector using:

|F₁|sin(θ) = F⊥

Question 5

What is the Cross Product formula?

Answer 5

A X B =

{(AᵧB₂ - A₂Bᵧ)i + (A₂Bₓ - AₓB₂)j + (AₓBᵧ - AᵧBₓ)k}

Question 6

What is the Cross Product used for?

Answer 6

Cross Product is used to calculate moments about a particular point in 3D. Where:

Mₒ = r̂ X F (where position vector is always first and is taken from the point we are taking the Moment about to any point on the force vector.

Resultant vector product produces a Moment about each axis:

Mₒ = {xi + yj + zk}

Question 7

How is a Moment taken about any arbitrary axis ?

Answer 7

Follow our Moment Vector process:

Μₒ = r̂ X F

Then we dot our Mₒ value to the unit vector of our desired direction:

Mₐ = Mₒ ⋅ ûₐ , since we’ve used dot product our new force is a scalar.