Table of contents |
2 Complex plane 3 Three dimensions 4 Orthogonal matrices 5 Relativity |
The magnitude of the original vector is
and the magnitude of the rotated vector isThen z can be rotated counterclockwise by an angle θ by pre-multiplying it with (see Euler's formula, §2), viz.
In ordinary three dimensional space, a coordinate rotation can be described by means of Eulerian angles. It can also be described by means of quaternions (see quaternions and spatial rotation), an approach which is similar to the use of vector calculus.
Another way is to multiply by a matrix M, which will rotate by an angle around a unit vector R:
This matrix is derived from the following vector algebraic equation:
Then, applying cyclic permutations to x, y, and z () the three resulting equations can be converted to similar ones for . It can thus be verified that
which shows that u is resolved (see Gram-Schmidt process) into a parallel and a perpendicular component (to R). The parallel component does not rotate, only the perpendicular component does rotate, and this rotation is similar to a two dimensional rotation, except that instead of x and y axes, there are and axes, both of which are perpendicular to R.
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. Orthogonal matrices are the real-valued version of unitary matrices.
In special relativity a Lorenzian coordinate rotation which rotates the time axis is called a boost, and, instead of spatial distance, the interval between any two points remains invariant. Lorentzian coordinate rotations which do not rotate the time axis are three dimensional spatial rotations. See Lorentz transformation, Lorentz group.