Coordinate rotation
In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant undr the transformation. In other words, it is an isometry - note that there are isometries other than rotations, though.
Two dimensions
In two dimensions, a counterclockwise coordinate rotation from a coordinate system  to a system  can be described by
-
 .
Then the magnitude of the vector (x,y) is the same as the magnitude of vector (x',y').
Proof
The magnitude of the original vector is
-
and the magnitude of the rotated vector is
-
Expand the squared binomials,
-
-
-
-
Which is the same as the original magnitude.
Complex plane
A complex number can be seen as a two-dimensional vector in the complex plane, with its tail at the origin and its head given by the complex number. Let
be such a complex number. Its real component is the abscissa and its imaginary component its ordinate.
Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with  (see Euler's formula, §2), viz.
This can be seen to correspond to the rotation described in § 1.
Three dimensions
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 below), 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:
Derivation
This matrix is derived from the following vector algebraic equation:
From this equation it is possible to calculate  by letting u' = x' and then dotting both sides of the equation by x, y, or z.
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
Equation (1) is in turn derived from
where  and  , as shown in the following diagram:
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.
Orthogonal matrices
The set of all matrices M('R,θ) described above is called the three-dimensional special orthogonal group: SO(3).
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. Orthogonal matrices are the real-valued version of unitary matrices.
Relativity
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.
Quaternions
Main article: Quaternions and spatial rotation
Quaternions are representing rotations and orientations in three dimensions. They are applied in computer graphics, control theory, signal processing and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.
See also
Referenced By
List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Lorentz group | Orthochronous | Quaternions and spatial rotation
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