Eric Lengyel: Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode..."

2023-07-15T06:36:24Z

Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode..."

New page

In geometric algebra, there are four ''commutator'' products defined as follows.

:$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$

:$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$

:$$[\mathbf a, \mathbf b]^{\Large\unicode{x27C7}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27C7}} \mathbf b - \mathbf b \mathbin{\unicode{x27C7}} \mathbf a\right)$$

:$$[\mathbf a, \mathbf b]^{\Large\unicode{x27C7}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27C7}} \mathbf b + \mathbf b \mathbin{\unicode{x27C7}} \mathbf a\right)$$

Commutators provide a way to formulate [[join and meet]] operations as well as [[Euclidean distances]] between different types of geometric objects. A commutator is also used to determine a new [[line]] containing the two closest points on a pair of skew [[lines]].

== See Also ==

* [[Join and meet]]
* [[Euclidean distance]]

Commutators - Revision history