Rotation

Let us try to define a rotation vector $\theta$ whose magnitude is the angle of the rotation, $\theta$, and whose direction is the axis of the rotation, in the sense determined by the right-hand grip rule. Is this a good vector? The short answer is, no. The problem is that the addition of rotations is not commutative, whereas vector addition is commuative. Figure 9 shows the effect of applying two successive $90^\circ$ rotations, one about $x$-axis, and the other about the $z$-axis, to a six-sided die. In the left-hand case, the $z$-rotation is applied before the $x$-rotation, and vice versa in the right-hand case. It can be seen that the die ends up in two completely different states. Clearly, the $z$-rotation plus the $x$-rotation does not equal the $x$-rotation plus the $z$-rotation. This non-commuting algebra cannot be represented by vectors. So, although rotations have a well-defined magnitude and direction, they are not vector quantities.

Figure 9:

But, this is not quite the end of the story. Suppose that we take a general vector $\bf a$ and rotate it about the $z$-axis by a small angle $\delta \theta_z$. This is equivalent to rotating the basis about the $z$-axis by $-\delta\theta_z$. According to Eqs. (10)-(12), we have

$${\bf a}' \simeq {\bf a} +\delta\theta_z {\bf e}_z\times {\bf a},$$ (39)

where use has been made of the small angle expansions $\sin\theta \simeq \theta$ and $\cos\theta\simeq 1$. The above equation can easily be generalized to allow small rotations about the $x$- and $y$-axes by $\delta \theta_x$ and $\delta\theta_y$, respectively. We find that

$${\bf a}' \simeq {\bf a} + \delta \mbox{\boldmath$\theta$}\times {\bf a},$$ (40)

where

$$\delta\mbox{\boldmath$\theta$} = \delta\theta_x {\bf e}_x + \delta\theta_y {\bf e}_y + \delta\theta_z {\bf e}_z.$$ (41)

Clearly, we can define a rotation vector $\delta$$\theta$, but it only works for small angle rotations (i.e., sufficiently small that the small angle expansions of sine and cosine are good). According to the above equation, a small $z$-rotation plus a small $x$-rotation is (approximately) equal to the two rotations applied in the opposite order. The fact that infinitesimal rotation is a vector implies that angular velocity,

$$\mbox{\boldmath$\omega$} = \lim_{\delta t\rightarrow 0} \frac{\delta \mbox{\boldmath$\theta$} }{\delta t},$$ (42)

must be a vector as well. Also, if ${\bf a}'$ is interpreted as ${\bf a}(t+\delta t)$ in the above equation then it is clear that the equation of motion of a vector precessing about the origin with angular velocity $\omega$ is

$$\frac{d {\bf a}}{dt} = \mbox{\boldmath$\omega$}\times {\bf a}.$$ (43)