From Vectors to Geometric Algebra

TL;DR

This paper introduces geometric algebra as a natural extension of vectors, providing algebraic tools that encode geometric theorems like Pythagoras and enable synthetic proofs across various dimensions and geometries.

Contribution

It presents a comprehensive review of vector addition and introduces a multiplication operation that encapsulates Euclidean geometry, extending applicability to multiple dimensions and geometries.

Findings

Algebraic encoding of the Pythagorean theorem

Replacement of synthetic proofs with algebraic methods

Applicability in Euclidean and non-Euclidean geometries

Abstract

Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous Pythagorean theorem. Synthetic proofs of theorems in Euclidean geometry can then be replaced by powerful algebraic proofs. Whereas we largely limit our attention to 2 and 3 dimensions, geometric algebra is applicable in any number of dimensions, and in both Euclidean and non-Euclidean geometries.