The geometric associative algebras of Euclidean space

We deduce two associative algebra structures arising from the homogeneity and isotropy of three dimensional space with an Euclidean geometry. These are the Clifford algebrasCl(3,0) andCl(0,3). We define a bivector as the geometric product of two vectors, a definition that differs from the usual. There is a choice of whether the bivectors are constructed tail to tail or head to head leading respectively to a positive definite or negative definite Euclidean metric. The origin of the two metric choices is not identified in the usual approach. Thus we arrive at a definite geometric answer to the question of Crumeyrolle, “What is a Bivector?”.