Vectors

Vector3 AtoB = B - A

Vectors will be written inside []. Ex [4,7,9].
- The zero vector of any dimension will use a boldface zero as variable. 0 = [0,0,..,0].
- To negate a vector we negate each component of the vector. [1,2,3] => [-1,-2,-3].
- The length of a vector v is written as ||v||. The length is the square root of the sum of the squares of the components of the vector. Ex v = [x,y,z]. ||v|| = sqrt(x²+y²+z²).
A unit vector is a vector that has the magnitude of 1. To normalize a vector, divide the vector by its magnitude. v_norm = v / ||v||.

Vector Dot Product

The dot product of two vectors is the sum of the products respective components, this results in a scalar. a·b = a₁b₁, ... , a_nb_n.
The dot product is equal to the product of the magnitude of the vectors and the cosine of the angle between the vectors. a·b = ||a|| ||b|| cos θ.
The angle between two vectors is. θ = acos(a·b / ||a|| ||b|| )
- If both vectors is unit vectors the divide is not needed so it's θ = acos(a·b)
- If we only care about the relative orientation about the vectors the acos can be skipped and only look at the sign of the dot product.
  - + : The vectors are in the same direction ( θ < 90°).
  - 0 : The vectors are perpendicular.
  - - : In opposite directions ( 90° < θ < 180°)

Projection one vector onto another

- Given two vectors v and n, we can separate v into two values, v_|| and v_t. They are parallel and perpendicular to n so that v = v_|| + v_t
- v_|| is the result of projecting v onto n.
- v_|| = n (v·n / ||n||²).

Vector cross product

Only works on 3D vectors.
It gives a vector that is perpendicular to the two original vectors.
The length of the vector is equal to the product of the magnitudes of a and b and the sine of the angle
Cross product use × as symbol.
a×b = -(b×a)
a×b = [a_yb_z-a_zb_y , a_zb_x - a_xb_z , a_xb_y - a_yb_x]

Reference

Practical applications of the dot product - 2017

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