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. vnorm = 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, ... , anbn.

  • 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 vt. They are parallel and perpendicular to n so that v = v|| + vt

    • 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 = [aybz-azby , azbx - axbz , axby - aybx]


Practical applications of the dot product - 2017

A faster quaternion-vector multiplication - 2013

Problem Solving Using Vectors - 2013

3D Math Primer for Game Programmers (Vector Operations) - 2011

Reflect and Refract Functions

Vector maths – a primer for games programmers

Linear algebra for game developers ~ part 1