Clifford Algebra - A Unified Language for Geometric Operations

Abstract: In this paper the Clifford Algebra is introduced and proposed as analternative to Gibbs' vector algebra as a unifying language for geometricoperations on vectors. Firstly, the algebra is constructed using a quotientof the tensor algebra and then its most important properties are proved,including how it enables division between vectors and how it is connected tothe exterior algebra. Further, the Clifford algebra is shown to naturallyembody the complex numbers and quaternions, whereupon its strength indescribing rotations is highlighted. Moreover, the wedge product, is shown asa way to generalize the cross product and reveal the true nature ofpseudovectors as bivectors. Lastly, we show how replacing the cross productwith the wedge product, within the Clifford algebra, naturally leads tosimplifying Maxwell's equations to a single equation.