Geometric Algebra.
This is an introduction to geometric algebra, an alternative to traditional
vector algebra that expands on it in two ways:
- In addition to scalars and vectors, it defines new objects representing
subspaces of any dimension.- 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.
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