Some Identities involving Two Sets of Basis Vectors and the Metric and Mixed Matrices

TL;DR

This paper explores mathematical relations between two sets of basis vectors in n-dimensional space, focusing on their metric and mixed matrices, with specific applications to 2D and 3D spaces using computer algebra.

Contribution

It provides explicit relations between basis vector lengths, angles, and matrices, extending understanding of geometric properties in higher dimensions.

Findings

Derived relations between metric and mixed matrices

Analyzed geometric implications in 2D and 3D spaces

Utilized computer algebra for simplifying complex expressions

Abstract

Given two sets of basis vectors in n-dimensional space, there exists a relation between their lengths and mutual angles, expressed as relations between the two metric matrices and the mixed matrix. In this paper these relations are given, and their consequences for 2-dimensional and 3-dimensional space are investigated, using a computer algebra program for simplifying expressions.