Commutators

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

$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde b} - \mathbf b \mathbin{\unicode{x27D1}} \mathbf{\tilde a}\right)$$
$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde b} + \mathbf b \mathbin{\unicode{x27D1}} \mathbf{\tilde a}\right)$$
$$[\mathbf a, \mathbf b]^{\Large\unicode{x27C7}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{b}} - \mathbf b \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{a}}\right)$$
$$[\mathbf a, \mathbf b]^{\Large\unicode{x27C7}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{b}} + \mathbf b \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{a}}\right)$$

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