Editing Cross product (section)

=== Commutator product ===
{{Main|Geometric algebra#Extensions of the inner and exterior products|Cross product#Cross product and handedness|Cross product#Lie algebra}}

Interpreting the three-dimensional [[vector space]] of the algebra as the [[bivector|2-vector]] (not the 1-vector) [[Graded vector space|subalgebra]] of the three-dimensional geometric algebra, where <math>\mathbf{i} = \mathbf{e_2} \mathbf{e_3}</math>, <math>\mathbf{j} = \mathbf{e_1} \mathbf{e_3}</math>, and <math>\mathbf{k} = \mathbf{e_1} \mathbf{e_2}</math>, the cross product corresponds exactly to the [[geometric algebra#Extensions of the inner and exterior products|commutator product]] in geometric algebra and both use the same symbol <math>\times</math>. The commutator product is defined for 2-vectors <math>A</math> and <math>B</math> in geometric algebra as:

: <math>A \times B = \tfrac{1}{2}(AB - BA),</math>

where <math>AB</math> is the geometric product.<ref>{{cite book|title=Understanding Geometric Algebra for Electromagnetic Theory|year=2011|last1=Arthur|first1=John W.|page=49|isbn=978-0470941638|publisher=[[IEEE Press]]|url=https://books.google.com/books?id=rxGCaDvBCoAC}}</ref>

The commutator product could be generalised to arbitrary [[multivector#Geometric algebra|multivectors]] in three dimensions, which results in a multivector consisting of only elements of [[Graded vector space|grades]] 1 (1-vectors/[[#Handedness|true vectors]]) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the [[Geometric algebra#Extensions of the inner and exterior products|left and right contractions]] in geometric algebra. The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the [[vector triple product]] of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the [[Sign (mathematics)#Sign of a direction|negative]] of the [[vector triple product]] of the same three true vectors in vector algebra.

Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. Just as the commutator product/cross product of 2-vectors in three dimensions [[#Lie algebra|correspond to the simplest Lie algebra]], the 2-vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the Lie algebras.<ref>{{cite book|title=Geometric Algebra for Physicists|year=2003|last1=Doran|first1=Chris|last2=Lasenby|first2=Anthony|pages=401–408|isbn=978-0521715959|publisher=[[Cambridge University Press]]|url=https://books.google.com/books?id=VW4yt0WHdjoC}}</ref> Also as in three dimensions, the commutator product could be further generalised to arbitrary multivectors.