Vectors

From modwiki

A vector is a mathematical quantity having both magnitude and direction. Common examples of quantities with only magnitude are mass(200kg), temperature(273 degrees Kelvin), and speed (15m/s) (not to be confused with velocity). These are also referred to as scalar quantities, because they are described with magnitude only. Vector quantities are those which are described with both magnitude and direction. Common examples are velocity (15m/s northeast) and force (150nt at 45 degrees).

A 2d vector a can be desribed by the equation:

 a = ix + jy

Where i is the magnitude of the component in the x plane, and j of the y plane.

A 3d vector is described in the same manner, but with 3 components:

 a = ix + jy + kz

With k being the magnitude of the z-plane component.

These three components combine to form a vector, representing all three components. For example, for a vector with the equation:

 a = 1 + 2x + -3z

The vector could be represented as a ray from the origin (0,0,0) through the point comprised of those three components. (1,2,-3)

The magnitude (also called modulus) |a| of the new resultant vector can be found using the Pythagorean Theorem:

 |a|² = x² + y² + z²
 |a|  = √(x² + y² + z²)

Normalizing a vector will result in a normal (or unit) vector, where the magnitude of the vector is equal to 1. This, in effect, describes only the direction of the vector.

To find a normalized vector â of vector a, divide the vector by it's modulus:

       a
 â =  ---
      |a|

Or

        x
 â.x = ---
       |a|

And so on for y,z,etc.

Since a normalized vector has length of 1,

 √(â.x² + â.y² + â.z²) = |â| = 1

For more information about vectors, see GameDev.net - Vectors and Matrices: A Primer

For a summary of vectors in the Doom 3 engine and functions they use, see Vectors.h