The Cross Products of M Vectors in N-dimensional Spaces and Their Geometric Significance

TL;DR

This paper generalizes the concept of the cross product to any number of vectors in high-dimensional spaces with various metric matrices, revealing its geometric significance and volume interpretation.

Contribution

It introduces a universal definition of the cross product for multiple vectors in n-dimensional spaces with arbitrary metric matrices, extending classical concepts.

Findings

Defines the cross product for m vectors in n-dimensional spaces with real symmetric or Hermitian metric matrices.

Shows the length of the cross product equals the volume of the spanned parallel polyhedron.

Demonstrates the cross product's components relate to volume projections in different directions.

Abstract

In textbooks and historical literature, the cross product has been defined only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product of only two vectors has been defined only in the high dimensional Euclidean space whose metric matrix is the unit matrix. Nobody has given a universal definition for any number of vectors in high dimensional spaces whose metric matrices are the unit matrices. In fact, we can also define the cross product of m vectors in an n-dimensional space, where n and m can take any positive integers larger than 1 and m must not be larger than n. In this paper, we give the definition of the cross product of m vectors in n-dimensional spaces whose metric matrices are any real symmetric or Hermitian matrices, and put forward two theorems related to matrices, so as to perfectly explain the geometric meaning of the cross product of vectors in high dimensional spaces. Furthermore, the length of the cross product represents the m-dimensional volume of the parallel polyhedron spanned by the m vectors, and the absolute value of each component of the cross product represents each component of the volume in different directions. Specially, in the high dimensional Euclidean and unitary space where the metric matrix is the unit matrix, this decomposition of the volume still satisfies the Pythagorean theorem, and the cross product of n vectors in an n-dimensional space is the determinant of the square matrix which is formed with these n vectors as row or column vectors. We also explain the geometric meaning of the determinants of the metric matrices and their sub-matrices, which is also useful for understanding the invariant volume elements in high dimensional spaces and their subspaces in the differential geometry.