Generalised vector products in threedimensional geometry
Abstract
In threedimensional 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 threedimensional 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 threedimensional affine space and the related twodimensional projective plane, and show an example of its applications to relativistic geometry.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 DOI:
 10.48550/arXiv.1903.08330
 arXiv:
 arXiv:1903.08330
 Bibcode:
 2019arXiv190308330N
 Keywords:

 Mathematics  Metric Geometry;
 51N10;
 51N15;
 15A63
 EPrint:
 33 pages