Connections between matrix products for 3-vectors and geometric algebra

Abstract

The geometric product represents a core concept for establishing geometric algebras and in case of vectors matches the formal sum of their inner product and their wedge product. The geometric product is reconsidered for the case of 3-vectors by means of usual matrix algebra in this article. Therefore, a symmetric matrix product and an antisymmetric matrix product are introduced, whose matrix sum yields a third product that renders the information of a vector pair’s geometric product in terms of a matrix associated to the vectors. The three matrices – that correspond to inner product, wedge product, and geometric product, respectively – are named wheel product, curl product, and full product. The observation about the structural correspondence of the geometric product with matrix theory may be used for future practical computations and unveils connections of geometric algebra with related disciplines.