Rotational coordinate transformations
Consider a conventional righthanded Cartesian coordinate system, , , .
Suppose that we transform to a new coordinate system, , , , that is obtained
from the , , system by
rotating the coordinate axes through an angle
about the axis. See Figure A.1.
Let the coordinates of a general point be
in the first coordinate system, and
in the second. According to simple trigonometry, these two sets of coordinates are related to one another via the transformation:
When expressed in matrix form, this transformation becomes

(A.89) 
The reverse transformation is accomplished by rotating the coordinate axes through an angle about
the axis:

(A.90) 
It follows that the matrix appearing in Equation (A.89) is the inverse of that appearing in Equation (A.90),
and vice versa.
However, because these two matrices are clearly also the transposes of one another, we deduce that both matrices are unitary.
In fact, it is easily demonstrated that all rotation matrices must be unitary; otherwise they would not
preserve the lengths of the vectors on which they act.
Figure: A.1
Rotation of the coordinate axes about the axis.

A rotation through an angle about the axis transforms the , , coordinate system into the , ,
system, where, by analogy with the previous analysis,

(A.91) 
Thus, from Equations (A.89) and (A.91), a rotation through an angle about the axis, followed by a rotation through an angle
about the axis, transforms the , , coordinate system into the , , system, where

(A.92) 