Introduction

Geometric algebra can simplify many rules and exceptions that are commonly found in vector algebra. As an algebra, it is a tool that allows one to express mathematical operations over certain entities. It is widely known that 2D dilations and rotations are more easily described with complex numbers, and to a lesser extent that quaternions are preferred for dealing with 3D rotations. It turns out that complex numbers and quaternions are sub-algebras of geometric algebra. To be exact, the term geometric algebra refers to a Clifford algebra of a vector space over the field of real numbers.

It uses a more complex structure than vector algebra, where vectors are defined as simple arrays of numbers that have no meaning other than being a list of coordinates. However, what we traditionally consider as vectors are in reality a little more complicated than that. For example, one may find that vectors that result from a cross-product do not follow the same invariance laws than vectors do in physics, and label them as pseudovectors. Geometric algebra accounts for this dissimilarity, describing the cross-product of two vectors as bivectors. The cross-product itself is substituted by another product, called the outer product, and is also defined in two dimensions without requiring a 3D framework. By considering vectors as mathematical entities that possess a particular structure, many theorems and laws appear more natural.