# Generalised vector products in three-dimensional geometry

## TL;DR

This paper extends classical three-dimensional vector calculus to arbitrary metrical geometries, generalizing key results and applying them to rational and relativistic geometries in algebraic and finite field settings.

## Contribution

It introduces a generalized vector product framework applicable to any three-dimensional metrical geometry, including finite fields, and connects it to rational and relativistic geometries.

## Key findings

- Generalized vector products for arbitrary metrical geometries.
- Application to rational trigonometry in 3D and projective planes.
- Example application to relativistic geometry.

## Abstract

In three-dimensional Euclidean geometry, the scalar product produces a number associated to two vectors, while the vector product computes a vector perpendicular to them. These are key tools of physics, chemistry and engineering and supported by a rich vector calculus of 18th and 19th century results. This paper extends this calculus to arbitrary metrical geometries on three-dimensional space, generalising key results of Lagrange, Jacobi, Binet and Cauchy in a purely algebraic setting which applies also to general fields, including finite fields. We will then apply these vector theorems to set up the basic framework of rational trigonometry in the three-dimensional affine space and the related two-dimensional projective plane, and show an example of its applications to relativistic geometry.