4.1. Introduction Points, Lines, Planes, and Hyperplanes
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.
-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.
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.