Rotational coordinate transformations

Consider a conventional right-handed Cartesian coordinate system, $x$, $y$, $z$. Suppose that we transform to a new coordinate system, $x'$, $y'$, $z'$, that is obtained from the $x$, $y$, $z$ system by rotating the coordinate axes through an angle $\theta$ about the $z$-axis. See Figure A.1. Let the coordinates of a general point $P$ be $(x,\,y,\,z)$ in the first coordinate system, and $(x',\, y',\, z')$ in the second. According to simple trigonometry, these two sets of coordinates are related to one another via the transformation:

$$x' = \cos\theta \,\,x+ \sin \theta\,\,y,$$ (A.86)

$$y' = -\sin\theta\,\,x + \cos\theta\,\,y,$$ (A.87)

$$z' = z.$$ (A.88)

When expressed in matrix form, this transformation becomes

$$\left(\begin{array}{c}x'\\ [0.5ex]y'\\ [0.5ex]z'\end{array}\right... ...d{array}\right)\left(\begin{array}{c}x\\ [0.5ex]y\\ [0.5ex]z\end{array}\right).$$ (A.89)

The reverse transformation is accomplished by rotating the coordinate axes through an angle $-\theta$ about the $z'$-axis:

$$\left(\begin{array}{c}x\\ [0.5ex]y\\ [0.5ex]z\end{array}\right)= ... ...rray}\right)\left(\begin{array}{c}x'\\ [0.5ex]y'\\ [0.5ex]z'\end{array}\right).$$ (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 $z$-axis.

A rotation through an angle $\phi$ about the $x'$-axis transforms the $x'$, $y'$, $z'$ coordinate system into the $x''$, $y''$, $z''$ system, where, by analogy with the previous analysis,

$$\left(\begin{array}{c}x''\\ [0.5ex]y''\\ [0.5ex]z''\end{array}\ri... ...rray}\right)\left(\begin{array}{c}x'\\ [0.5ex]y'\\ [0.5ex]z'\end{array}\right).$$ (A.91)

Thus, from Equations (A.89) and (A.91), a rotation through an angle $\theta$ about the $z$-axis, followed by a rotation through an angle $\phi$ about the $x'$-axis, transforms the $x$, $y$, $z$ coordinate system into the $x''$, $y''$, $z''$ system, where

$$\left(\begin{array}{c}x''\\ [0.5ex]y''\\ [0.5ex]z''\end{array}\ri... ...d{array}\right)\left(\begin{array}{c}x\\ [0.5ex]y\\ [0.5ex]z\end{array}\right).$$ (A.92)