So far we have referred to -dimensional space and the vectors contained within such a space. When discussing the properties of vectors such as magnitude, direction, angles between vectors, and vectors normal to other vectors, we have briefly described some of the geometry behind -dimensional space. In addition to vectors, -dimensional space also contains points, lines, planes, and hyperplanes which have 0, 1, 2, and 3 or greater dimensions respectively.

In order to work with various geometric objects we need a way to specify them. In elementary algebra, we dealt with the 2-dimensional Cartesian plane. Points were specified by two coordinates, for example . This specification is actually two equations for two lines: the line and the line . The intersubsection of these two lines is the point . The specification of is not unique in that we could have used any two lines that intersect at this point to define it as shown in Figure 4-1.
Object could not be loaded.
Figure 4-1. Multiple Definitions of a Point
To specify an object of dimensions in a space of dimensions you need equations. So in dimensional space, we need two equations to specify a point since it has dimensions and . We need one equation to specify a line in 2D since . In 3-dimensional space 1 equation defines a plane () while 2 equations are required to define a line ().

The rest of the subsections in this chapter will be mostly concerned with how to represent geometric objects using various mathematical forms.