History is written by the winners - and so it is with vector algebra. Around the turn of the 19th century, a battle was under way to construct a vector formalism to describe 3 dimensional Euclidean space. On one side there was Josiah Gibbs and Oliver Heaviside and the other side was William Clifford. The vector algebra of Gibbs and Heaviside prevailed and continues to dominate calculus classes today. The aim of this talk is to argue that Clifford's system, known as a Clifford algebra (also known as geometric algebra) is a superior formalism for such a task. In this talk, some basic definitions and constructions of Clifford algebras are presented, as well as some applications.