Editing Rotation (mathematics) (section)

====Quaternions====
{{Main|Quaternions and spatial rotation}}

Unit [[quaternion]]s, or ''[[versor]]s'', are in some ways the least intuitive representation of three-dimensional rotations. They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.{{Citation needed|date=July 2010}}

A versor (also called a ''rotation quaternion'') consists of four real numbers, constrained so the [[normed vector space|norm]] of the quaternion is 1. This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:

:<math> \mathbf{x'} = \mathbf{qxq}^{-1},</math>

where {{math|'''q'''}} is the versor, {{math|'''q'''<sup>−1</sup>}} is its [[multiplicative inverse|inverse]], and {{math|'''x'''}} is the vector treated as a quaternion with zero [[quaternion#Scalar and vector parts|scalar part]]. The quaternion can be related to the rotation vector form of the axis angle rotation by the [[exponential map (Lie theory)|exponential map]] over the quaternions,

:<math> \mathbf{q}  = e^{\mathbf{v}/2},</math>

where {{math|'''v'''}} is the rotation vector treated as a quaternion.

A single multiplication by a versor, [[left and right (algebra)|either left or right]], is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two ''different'' unit quaternions.