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Euclidean Clifford Algebra

Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of \emph{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 \emph{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 \emph{contraction operators} on $\bigwedge V,$ and the concept of euclidean \emph{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 develope a skill on the subject, preparing himself for the reading of the following papers in this series.