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Let us try to define a rotation vector $ \theta $ whose magnitude is the angle of the rotation, $ \theta $ , and whose direction is parallel to the axis of rotation, in the sense determined by a right-hand circulation rule. Unfortunately, this is not a good vector. The problem is that the addition of rotations is not commutative, whereas vector addition is commuative. Figure A.11 shows the effect of applying two successive $ 90^\circ $ rotations, one about $ Ox$ , and the other about the $ Oz$ , to a standard 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. In other words, 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 A.11: Effect of successive rotations about perpendicular axes on a six-sided die.
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But, this is not quite the end of the story. Suppose that we take a general vector $ \bf a$ , and rotate it about $ Oz$ by a small angle $ \delta \theta_z$ . This is equivalent to rotating the coordinate axes about $ Oz$ by $ -\delta\theta_z$ . According to Equations (A.20)-(A.22), we have

$\displaystyle {\bf a}' \simeq {\bf a} +\delta\theta_z \,{\bf e}_z\times {\bf a},$ (A.48)

where use has been made of the small angle approximations $ \sin\theta \simeq \theta$ and $ \cos\theta\simeq 1$ . The previous equation can easily be generalized to allow small rotations about $ Ox$ and $ Oy$ by $ \delta \theta_x$ and $ \delta\theta_y$ , respectively. We find that

$\displaystyle {\bf a}' \simeq {\bf a} + \delta$   $\displaystyle \mbox{\boldmath$\theta$}$$\displaystyle \times {\bf a},$ (A.49)


$\displaystyle \delta$$\displaystyle \mbox{\boldmath$\theta$}$$\displaystyle = \delta\theta_x \,{\bf e}_x + \delta\theta_y \,{\bf e}_y + \delta\theta_z \,{\bf e}_z.$ (A.50)

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 approximations of sine and cosine are good). According to the previous 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,

$\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle = \lim_{\delta t\rightarrow 0} \frac{\delta \mbox{\boldmath$\theta$} }{\delta t},$ (A.51)

must be a vector as well. Also, if $ {\bf a}'$ is interpreted as $ {\bf a}(t+\delta t)$ in Equation (A.49) then it follows that the equation of motion of a vector that precesses about the origin with some angular velocity $ \omega$ is

$\displaystyle \frac{d {\bf a}}{dt} =$   $\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \times {\bf a}.$ (A.52)

next up previous
Next: Scalar Triple Product Up: Vectors and Vector Fields Previous: Vector Product
Richard Fitzpatrick 2016-03-31