When Does a Cross Product on R^{n} Exist?

TL;DR

This paper provides an accessible, elementary proof that the cross product with familiar properties exists only in three and seven-dimensional real spaces, clarifying the classical result in linear algebra.

Contribution

It offers a new constructive proof of the classical fact that the cross product's properties extend only to R^3 and R^7, suitable for undergraduate education.

Findings

Cross product exists only in R^3 and R^7

Elementary proof accessible to undergraduates

Clarifies classical mathematical result

Abstract

It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what many readers may not be aware of is that the familiar properties of the cross product in three space can only be extended to R^{7}. The intention of this article is to provide a new constructive elementary proof (i.e. could be included in an linear algebra undergraduate text) of this classical result which is accessible to a wide audience.