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2 Complex plane 3 Three dimensions 4 Orthogonal matrices 5 Relativity |
In two dimensions, a counterclockwise coordinate rotation from a coordinate system to a system can be described by
The magnitude of the original vector is
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
Then 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.
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.
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
and the magnitude of the rotated vector is
Expand the squared binomials,
Which is the same as the original magnitude.Complex plane
be such a complex number. Its real component is the abscissa and its imaginary component its ordinate.
This can be seen to correspond to the rotation described in § 1.Three dimensions
Derivation
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