TITLE:
Geometric Analogy and Products of Vectors in n Dimensions
AUTHORS:
Leonardo Simal Moreira
KEYWORDS:
Cross Product; Space IRn; Determinants; Geometric Analogy; Eckman’s Product
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.3 No.1,
March
27,
2013
ABSTRACT: 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, basedon known properties of determinants.