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Geometric Analogy and Products of Vectors in n Dimensions

DOI: 10.4236/alamt.2013.31001, PP. 1-6

Keywords: Cross Product, Space IRn, Determinants, Geometric Analogy, Eckman’s Product

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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, based on known properties of determinants.

References

[1]  B. Eckmann, “Stetige L?sungen Linearer Gleichungssysteme,” Commentarii Mathematici Helvetici, Vol. 15, 1943, pp. 318-339. doi:10.1007/BF02565648
[2]  N. Efimov, “Elementos de Geometria Analítica,” Cultura Brasileira, S?o Paulo, 1972.
[3]  A. Elduque, “Vector Cross Products,” Talk Presented at the Seminario Rubio de Francia of the Universidad de Zaragoza on April 1 2004.
[4]  S. Lipschutz and M. Lipson, “álgebra Linear,” Bookman, Porto Alegre, 2008.
[5]  R. Brown and A. Gray, “Vector Cross Products,” Commentarii Mathematici Helvetici, Vol. 42, 1967, pp. 222-236. doi:10.1007/BF02564418
[6]  A. Gray, “Vector Cross Products on Manifolds,” University of Maryland, College Park, 1968.
[7]  P. Gritzmann and V. Klee, “On the Complexity of Some Basic Problems in Computational Convexity II. Volume and Mixed Volumes,” In: T. Bisztriczky, P. McMuffen, R. Schneider and A. W. Weiss, Eds., Polytopes: Abstract, Convex and Computational, Kluwer, Dordrecht, 1994, p. 29.
[8]  D. M. Y. Sommerville, “An Introduction to the Geometry of n Dimensions,” Dover, New York, 1958, p. 124.

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