0.Math

Vectors: Representation

Vectors: Stereo View

Vectors: Scaling

Vectors: Addition

Vectors: Normalization

Length of vector

V =

[

x
y

]

|V| =

sqrt(x² + y²)

Vectors: Normalization

V' =

|V|

[

x/sqrt(x² + y²)
y/sqrt(x² + y²)

]

Vectors: Normalization

V =

[

4
3

]

|V| =

sqrt(3² + 4²) = 5

V' =

[

4/5
3/5

]

Vectors: Dot Product (scalar product)

V₁ =

[

x₁
y₁

]

V₂ =

[

x₂
y₂

]

Vectors: Dot Product (scalar product)

V₁ =

[

x₁
y₁

]

V₂ =

[

x₂
y₂

]

if |V₁| = |V₂| = 1

V₁^.V₂ = |V₁|^.|V₂|^.cos[theta] = x₁^.x₂ + y₁^.y₂

Vectors: Dot Product (scalar product)

V₁^.V₂ = ?

Vectors: Dot Product (scalar product)

V₁^.V₂ = 0

V₁^.V₂ = |V₁|²

Vectors: Dot Product (scalar product)

V₁^.V₂ = 0

V₁^.V₂ = |V₁|²

=> Dot Product indicates angle between vectors

On unit vectors, dot product gives length of projection of one vector on another.

Vectors: The Sign of the Dot Product

Vectors: Cross Product (Vector Product)

2 vectors define a plane

Vectors: Cross Product (Vector Product)

2 vectors define a plane

Cross product is perpendicular to that plane with length equal to area of parallelogram

Vectors: Cross Product (Vector Product)

2 vectors define a plane

Cross product is perpendicular to that plane with length equal to area of parallelogram

V₁ x V₁ = U |V₁|^.|V₂| sin[theta]

[

V_1yV_2z - V_1zV_2y
V_1zV_2x - V_1xV_2z
V_1xV_2y - V_1yV_2x

]

Matrix Multiplication

C = A^.B

C_ij=sum[k] ( a_ikb_kj )

Matrix Multiplication

C = A^.B

C_ij=sum[k] ( a_ikb_kj )

C_0,0 = a_0,0b_0,0 + a_0,1b_1,0

Matrix Multiplication

C = A^.B

C_ij=sum[k] ( a_ikb_kj )

C_0,0 = a_0,0b_0,0 + a_0,1b_1,0

A = I^.A (multiplicative identity)

where I = [ [ 1 0 ] [ 0 1 ] ]

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