Euclidean Clifford Algebra

Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of euclidean Clifford algebra $\mathcal{C\ell }(V,G_{E})$ for a given euclidean structure on $V$, i.e., a pair $(V,G_{E})$ where $G_{E} $ is a euclidean metric for $V$ (also called an euclidean scalar product). Our construction of $\mathcal{C\ell }(V,G_{E})$ has been designed to produce a powerful computational tool. We start introducing the concept of multivectors over $V$. These objects are elements of a linear space over the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two contraction operators on $\bigwedge V$, and the concept of euclidean interior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develop a skill on the subject, preparing himself for the reading of the following papers in this series.