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.
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2 Proof 3 Three dimensions 4 Orthogonal matrices 5 Relativity |
In two dimensions, a coordinate rotation from a coordinate system to a system can be described by
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:
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 are represented by orthogonal 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.Two dimensions
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.Three dimensions
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.
Equation (1) is in turn derived from
where and , as shown in the following diagram:Orthogonal matrices
Relativity