Cross product in N Dimensions - the doublewedge product

TL;DR

This paper introduces a novel extension of the cross product operator to N-dimensional space, addressing limitations of the traditional 3D version and facilitating calculations of moments in higher dimensions.

Contribution

It presents a new formulation of the cross product applicable in arbitrary N-dimensional spaces, distinct from exterior algebra, for easier computation of moments.

Findings

Extended cross product defined for N dimensions

Simplifies calculations of moments in higher dimensions

Addresses limitations of traditional 3D cross product

Abstract

The cross product frequently occurs in Physics and Engineering, since it has large applications in many contexts, e.g. for calculating angular momenta, torques, rotations, volumes etc. Though this mathematical operator is widely used, it is commonly expressed in a 3-D notation which gives rise to many paradoxes and difficulties. In fact, instead of other vector operators like scalar product, the cross product is defined just in 3-D space, it does not respect reflection rules and invokes the concept of "handedness". In this paper we are going to present an extension of cross product in an arbitrary number N of spatial Dimensions, different from the one adopted in the Exterior Algebra and explicitly designed for an easy calculus of moments.