Rotational coordinate transformations

When expressed in matrix form, this transformation becomes

The reverse transformation is accomplished by rotating the coordinate axes through an angle about the -axis:

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.

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

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) |