Geometric Analogy and Products of Vectors in *n* Dimensions

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.

Cite this paper

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.

L. Simal Moreira, "Geometric Analogy and Products of Vectors in

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.

[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.