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## Rotation

Let us try to define a rotation vector whose magnitude is the angle of the rotation, , 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 rotations, one about -axis, and the other about the -axis, to a six-sided die. In the left-hand case, the -rotation is applied before the -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 -rotation plus the -rotation does not equal the -rotation plus the -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.

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

 (39)

where use has been made of the small angle expansions and . The above equation can easily be generalized to allow small rotations about the - and -axes by and , respectively. We find that
 (40)

where
 (41)

Clearly, we can define a rotation vector , 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 -rotation plus a small -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,
 (42)

must be a vector as well. Also, if is interpreted as in the above equation then it is clear that the equation of motion of a vector precessing about the origin with angular velocity is
 (43)

Next: The scalar triple product Up: Vectors Previous: The vector product
Richard Fitzpatrick 2006-02-02