Geometric Algebra

TL;DR

This paper introduces geometric algebra, an extension of traditional vector algebra that includes new objects representing subspaces and a geometry-inspired product, aiming to establish it as the standard vector algebra system.

Contribution

It presents geometric algebra as a comprehensive, unified framework that generalizes vector algebra and highlights its historical development and potential advantages.

Findings

Geometric algebra includes objects for subspaces of any dimension.

It defines a product motivated by geometry applicable to any objects.

The system encompasses and extends traditional vector algebra.

Abstract

This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness. The notes are work in progress; I'll keep adding new topics as I learn them myself.