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 ALAMT  Vol.3 No.1 , March 2013
Geometric Analogy and Products of Vectors in n Dimensions
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.
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.
References

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[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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