Geometric algebra was introduced in the nineteenth century by the English mathematician William Kingdon Clifford (figure 2.1). Clifford appears to have been one of the small number of mathematicians at the time to be significantly influenced by Grassmann's work. Clifford introduced his geometric algebra by uniting the inner and outer products into a single geometric product. This is associative, like Grassmann's product, but has the crucial extra feature of being invertible, like Hamilton's quaternion algebra. Indeed, Clifford's original motivation was to unite Grassmann's and Hamilton's work into a single structure. In the mathematical literature one often sees this subject referred to as Clifford algebra. We have chosen to follow the example of David Hestenes, and many other modern researchers, by returning to Clifford's original choice of name – geometric algebra. One reason for this is that the first published definition of the geometric product was due to Grassmann, who introduced it in the second Ausdehnungslehre. It was Clifford, however, who realised the great potential of this product and who was responsible for advancing the subject.

In this chapter we introduce the basics of geometric algebra in two and three dimensions in a way that is intended to appear natural and geometric, if somewhat informal. A more formal, axiomatic approach is delayed until chapter 4, where geometric algebra is defined in arbitrary dimensions.