Editing Geometric algebra (section)

=== Versor ===
A {{tmath|1= k }}-versor is a multivector that can be expressed as the geometric product of <math>k</math> invertible vectors.{{efn|"reviving and generalizing somewhat a term from hamilton's quaternion calculus which has fallen into disuse" Hestenes defined a {{tmath|1= k }}-versor as a multivector which can be factored into a product of <math>k</math> vectors.{{sfn|ps=|Hestenes|Sobczyk|1984|p=103}}}}{{sfn|ps=|Dorst|Fontijne|Mann|2007|p=204}} Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refer to Dorst.{{sfn|ps=|Dorst|Fontijne|Mann|2007|pp=177–182}}

Some authors use the term "versor product" to refer to the frequently occurring case where an operand is "sandwiched" between operators. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. These outermorphisms have a particularly simple algebraic form.{{efn|Only the outermorphisms of linear transformations that respect the bilinear form fit this description; outermorphisms are not in general expressible in terms of the algebraic operations.}} Specifically, a mapping of vectors of the form
: <math> V \to V : a \mapsto RaR^{-1}</math> extends to the outermorphism <math>\mathcal{G}(V) \to \mathcal{G}(V) : A \mapsto RAR^{-1}.</math>

Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.

By the [[Cartan–Dieudonné theorem]] we have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors.

In group terms, for a real, non-degenerate {{tmath|1= \mathcal{G}(p,q) }}, having identified the group <math>\mathcal{G}^\times</math> as the group of all invertible elements of {{tmath|1= \mathcal{G} }}, Lundholm gives a proof that the "versor group" <math>\{ v_1 v_2 \cdots v_k \in \mathcal{G} \mid v_i \in V^\times\}</math> (the set of invertible versors) is equal to the Lipschitz group <math>\Gamma</math> ({{aka}} Clifford group, although Lundholm deprecates this usage).{{sfn|ps=|Lundholm|Svensson|2009|pp=58 ''et seq''}}