Geometric Algebra of the Plane

A 1-dimensional space has insufficient geometric structure to show what is going on, so we begin in two dimensions, taking two orthonormal basis vectors and . These satisfy the relations

and

The outer product represents the directed area element of the plane and we assume that , are chosen such that this has the conventional right-handed orientation. This completes the geometrically meaningful quantities that we can make from these basis vectors:

We now assemble a Clifford algebra from these quantities. An arbitrary linear sum over the four basis elements in (2.6) is called a multivector. In turn, given two multivectors A and B, we can form their sum S = A + B by adding the components:

By this definition of a linear sum we have done almost nothing - the power comes from the definition of the multiplication P = AB. In order to define this product, we have to be able to multiply the 4 geometric basis elements. Multiplication by a scalar is obvious. To form the products of the vectors we remember the definition , so that

Products involving the bivector are particularly important. Since the geometric product is associative, we have:

and

The only other product is the square of :

These results complete the definition of the product and enable, for example, the processes of addition and multiplication to be coded as computer functions. In principle, these definitions could be made an intrinsic part of a computer language, in the same way that complex number arithmetic is already intrinsic to some languages. To reinforce this point, it may be helpful to write out the product explicitly. We have,

where

Multivector addition and multiplication obey the associative and distributive laws, so that we have, as promised, the geometric algebra of the plane.

We emphasise the important features that have emerged in the course of this derivation.

• The geometric product of two parallel vectors is a scalar number - the product of their lengths.
• The geometric product of two perpendicular vectors is a bivector - the directed area formed by the vectors.
• Parallel vectors commute under the geometric product; perpendicular vectors anticommute.
• The bivector has the geometric effect of rotating the vectors in its own plane by clockwise when multiplying them on their left. It rotates vectors by anticlockwise when multiplying on their right. This can be used to define the orientation of and .
• The square of the bivector area is a scalar: .