Editing 3D rotation group (section)

==Length and angle==
Besides just preserving length, rotations also preserve the [[angle]]s between vectors. This follows from the fact that the standard [[dot product]] between two vectors '''u''' and '''v''' can be written purely in terms of length (see the [[law of cosines]]):
<math display="block">\mathbf{u} \cdot \mathbf{v} = \frac{1}{2} \left(\|\mathbf{u} + \mathbf{v}\|^2 - \|\mathbf{u}\|^2 - \|\mathbf{v}\|^2\right).</math>

It follows that every length-preserving linear transformation in <math>\R^3</math> preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on <math>\R^3</math>, which is equivalent to requiring them to preserve length. See [[classical group]] for a treatment of this more general approach, where {{math|SO(3)}} appears as a special case.