Vector cross product in n-dimensional vector space

TL;DR

This paper generalizes the vector cross product to odd-dimensional vector spaces by introducing a new cross term, extending its classical definitions beyond three and seven dimensions.

Contribution

It provides a novel generalized definition of the vector cross product applicable to all odd-dimensional spaces, validated by reduction to known cases.

Findings

Generalized VCP defined for odd n-dimensional spaces

Validation through reduction to 3D and 7D cases

Potential extension of VCP applications in higher dimensions

Abstract

The definition of vector cross product (VCP) introduced by Eckmann only exists in thethree- and the seven- dimensional vector space. In this paper, according to the orthogonal completeness, magnitude of basis vector cross product and all kinds of combinations of basis vector $\hat{e}_i$, the generalized definition of VCP in the odd n-dimensional vector space is given by introducing a cross term $X_{AB}$. In addition, the definition is validated by reducing the generalization definition to the fundamental three- and seven-dimensional vector space.