Geometric Analogy and Products of Vectors in <i>n</i> Dimensions
- 1 UniFOA—Centro Universitário de Volta Redonda, Volta Redonda, Brazil
Abstract
The cross product in Euclidean space IR 3 is an operation in which two vectors are associated to generate a third vector, also in space IR 3 . This product can be studied rewriting its basic equations in a matrix structure, more specifically in terms of determinants. Such a structure allows extending, for analogy, the ideas of the cross product for a type of the product of vectors in higher dimensions, through the systematic increase of the number of rows and columns in determinants that constitute the equations. So, in a n -dimensional space with Euclidean norm, we can associate n – 1 vectors and to obtain an n -th vector, with the same geometric characteristics of the product in three dimensions. This kind of operation is also a geometric interpretation of the product defined by Eckman [ 1 ]. The same analogies are also useful in the verification of algebraic properties of such products, based on known properties of determinants.
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