Mathematical Prerequisites - Introduction to 3D Game Programming with DirectX 9.0 - page 32

Summary

Vectors are used to model physical quantities that possess a magni-
tude and direction mathematically. Geometrically, we represent a
vector with a directed line segment. A vector is in standard position
when it is translated parallel to itself so that its tail coincides with
the origin of the coordinate system. A vector in standard position
can be described algebraically by specifying the coordinates of its
head.
We can use 4 4 matrices to represent transformations and 1 4
homogeneous vectors to describe points and vectors. The vector-
matrix product of a 1 4 row vector and a 4 4 transformation
matrix results in a new transformed 1 4 row vector. Several trans-
formation matrices can be combined into one transformation matrix
through matrix-matrix multiplication.

We use 4D homogeneous vectors to represent both vectors and
points. We specify 0 for the w component to denote a vector and 1

for the w 1, then we

have a vector ( x, y, z, w ) in homogeneous space that can be
mapped back to 3-space by dividing each component by w :

w component to denote a point. If w 0 and

xy zw xy z xy z
wwww www www

, , , , , , 1 , ,

Planes divide 3D space into two parts: a positive half-space, which
is the space in front of the plane, and a negative half-space, which is
the space behind the plane. Planes are useful for testing the locality
of points relative to them (in other words, what half-space a point
exists in relative to a particular plane).

Rays are described parametrically with an origin and direction vec-
tor. Rays are useful for modeling various physical quantities, such
as light rays and projectiles that approximately follow a linear path
such as bullets and rockets.