Editing 3D rotation group (section)

==Orthogonal and rotation matrices==
{{Main|Orthogonal matrix|Rotation matrix}}

Every rotation maps an [[orthonormal basis]] of <math>\R^3</math> to another orthonormal basis. Like any linear transformation of [[finite-dimensional]] vector spaces, a rotation can always be represented by a [[matrix (mathematics)|matrix]]. Let {{math|''R''}} be a given rotation. With respect to the [[standard basis]] {{math|'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>}} of <math>\R^3</math> the columns of {{math|''R''}} are given by {{math|(''R'''''e'''<sub>1</sub>, ''R'''''e'''<sub>2</sub>, ''R'''''e'''<sub>3</sub>)}}. Since the standard basis is orthonormal, and since {{math|''R''}} preserves angles and length, the columns of {{math|''R''}} form another orthonormal basis. This orthonormality condition can be expressed in the form

:<math>R^\mathsf{T}R = RR^\mathsf{T} = I,</math>

where {{math|''R''<sup>{{sans-serif|T}}</sup>}} denotes the [[transpose]] of {{math|''R''}} and {{mvar|I}} is the {{math|3 × 3}} [[identity matrix]]. Matrices for which this property holds are called [[orthogonal matrix|orthogonal matrices]]. The group of all {{math|3 × 3}} orthogonal matrices is denoted {{math|O(3)}}, and consists of all proper and improper rotations.

In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the [[determinant]] of the matrix is positive or negative. For an orthogonal matrix {{math|''R''}}, note that {{math|1=det ''R''<sup>{{sans-serif|T}}</sup> = det ''R''}} implies {{math|1=(det ''R'')<sup>2</sup> = 1}}, so that {{math|1=det ''R'' = ±1}}. The [[subgroup]] of orthogonal matrices with determinant {{math|+1}} is called the ''special [[orthogonal group]]'', denoted {{math|SO(3)}}.

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to [[matrix multiplication]], the rotation group is [[isomorphic]] to the special orthogonal group {{math|SO(3)}}.

[[Improper rotation]]s correspond to orthogonal matrices with determinant {{math|−1}}, and they do not form a group because the product of two improper rotations is a proper rotation.