Editing 3D rotation group (section)

{{Short description|Group of rotations in 3 dimensions}}
In [[classical mechanics|mechanics]] and [[geometry]], the '''3D rotation group''', often denoted '''[[special orthogonal group|SO]](3)''', is the [[group (mathematics)|group]] of all [[rotation]]s about the [[origin (mathematics)|origin]] of [[three-dimensional space|three-dimensional]] [[Euclidean space]] <math>\R^3</math> under the operation of [[function composition|composition]].<ref>Jacobson (2009), p. 34, Ex. 14.</ref>

By definition, a rotation about the origin is a transformation that preserves the origin, [[Euclidean distance]] (so it is an [[isometry]]), and [[orientation (vector space)|orientation]] (i.e., ''handedness'' of space). Composing two rotations results in another rotation, every rotation has a unique [[Inverse function|inverse]] rotation, and the [[identity map]] satisfies the definition of a rotation. Owing to the above properties (along composite rotations' [[associative property]]), the set of all rotations is a [[group (mathematics)|group]] under composition.

Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating ''R'' 90° in the x-y plane followed by ''S'' 90° in the y-z plane is not the same as ''S'' followed by ''R''), making the 3D rotation group a [[Non-abelian group|nonabelian group]]. Moreover, the rotation group has a natural structure as a [[manifold]] for which the group operations are [[smooth function|smoothly differentiable]], so it is in fact a [[Lie group]]. It is [[compact space|compact]] and has dimension 3.

Rotations are [[linear transformation]]s of <math>\R^3</math> and can therefore be represented by [[matrix (mathematics)|matrices]] once a [[basis of a vector space|basis]] of <math>\R^3</math> has been chosen. Specifically, if we choose an [[orthonormal basis]] of <math>\R^3</math>, every rotation is described by an [[orthogonal matrix|orthogonal 3 × 3 matrix]] (i.e., a 3 × 3 matrix with real entries which, when multiplied by its [[transpose]], results in the [[identity matrix]]) with [[determinant]] 1. The group SO(3) can therefore be identified with the group of these matrices under [[matrix multiplication]]. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).

The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its [[group representation|representation]]s are important in physics, where they give rise to the [[elementary particle]]s of integer [[Spin (physics)|spin]].