Editing Geometric algebra (section)

=== Rotations ===
[[File:GA planar rotations.svg|right|200px|thumb|A rotor that rotates vectors in a plane rotates vectors through angle {{tmath|1= \theta }}, that is <math>x \mapsto R_\theta x \widetilde R_\theta</math> is a rotation of <math>x</math> through angle {{tmath|1= \theta }}. The angle between <math>u</math> and <math>v</math> is {{tmath|1= \theta/2 }}. Similar interpretations are valid for a general multivector <math>X</math> instead of the vector {{tmath|1= x }}.{{sfn|ps=|Hestenes|2005}}]]

If we have a product of vectors <math>R = a_1a_2 \cdots a_r</math> then we denote the reverse as
: <math>\widetilde R = a_r\cdots a_2 a_1.</math>

As an example, assume that <math> R = ab </math> we get
: <math>R\widetilde R = abba = ab^2a = a^2b^2 = ba^2b = baab = \widetilde RR.</math>

Scaling <math>R</math> so that <math>R\widetilde R = 1</math> then
: <math>(Rv\widetilde R)^2 = Rv^{2}\widetilde R = v^2R\widetilde R = v^2 </math>
so <math>Rv\widetilde R</math> leaves the length of <math>v</math> unchanged. We can also show that
: <math>(Rv_1\widetilde R) \cdot (Rv_2\widetilde R) = v_1 \cdot v_2</math>
so the transformation <math>Rv\widetilde R</math> preserves both length and angle. It therefore can be identified as a rotation or rotoreflection; <math>R</math> is called a [[rotor (mathematics)|rotor]] if it is a [[proper rotation]] (as it is if it can be expressed as a product of an even number of vectors) and is an instance of what is known in GA as a ''[[versor]]''.

There is a general method for rotating a vector involving the formation of a multivector of the form <math> R = e^{-B \theta / 2} </math> that produces a rotation <math> \theta </math> in the [[Plane of rotation|plane]] and with the orientation defined by a {{tmath|1= 2 }}-blade {{tmath|1=  B }}.

Rotors are a generalization of quaternions to {{tmath|1= n }}-dimensional spaces.