Essential 3D math - Vector

Following math concepts are essential for further understanding.

A 3D vector is represented as [Vx, Vy,  Vz] where  Vx, Vy and Vz represent numbers in 3D cartesian space. Vectors are attributed with a direction represented by its head  and a magnitude computed as square root of the squared sums of its components.

Unary negation and scalar multiplication 

The  negation and multiplication is done on all the components.

For example consider a vector

v=[1,2,3] so  2*v results in [2,4,6]. Similarly -v results in [-1,-2,-3].

Addition and Subtraction

Vectors can be added or subtracted. Here the individual components are added or subtracted. 

For example consider two vectors  A[1,2,3] and B[4,5,6]. 

A+B = [(1+4), (2+5), (3+6)] = [5, 7, 9].

A-B =  [(1-4), (2-5), (3-6)] = [-3, 3, -3].

Graphically represented as below, Note that the direction has changed  when arguments are reversed.

Unary Vectors

also known as normalized vectors are represented as V^{^}  have magnitude of 1. 

For Example V = [3,2,1]. It's normalized vector is calculated as

[2,3,1]/sqrt(4+9+1) = [3/3.74, 2/3.74, 1/3.74]=[0.8, 0.53, 0,27]

Dot Product

Dot Product of  two vectors is equal to the product of their magnitudes and the cosine of the angle between them. The notation used is a.b where a and b are vectors. It's mathematically represented as

a ·b = |a| x |b| x cos(θ).

It's graphically represented as 

To compute angle between two unit vectors, mathematically it can be represented as

The dot product may be a positive  or a negative or a zero.

Cross Product

Cross Product of  two vectors is equal to the product of their magnitudes and the sine of the angle between them. The notation used is axb where a and b are vectors.

Mathematically it's represented as

Cross product yields a vector that is perpendicular to the plane of two vectors a and b. Graphically it's represented as

The cross product of two vector is equal to area of their parallelogram.