Geometric Analogy and Products of Vectors in n Dimensions


The cross product in Euclidean space IR3 is an operation in which two vectors are associated to generate a third vector, also in space IR3. 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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L. Simal Moreira, "Geometric Analogy and Products of Vectors in n Dimensions," Advances in Linear Algebra & Matrix Theory, Vol. 3 No. 1, 2013, pp. 1-6. doi: 10.4236/alamt.2013.31001.

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