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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 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 rotations, one about , and the other about the , to a standard 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. In other words, 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 by a small angle . This is equivalent to rotating the coordinate axes about by . According to Equations (A.20)-(A.22), we have

 (A.48)

where use has been made of the small angle approximations and . The previous equation can easily be generalized to allow small rotations about and by and , respectively. We find that

 (A.49)

where

 (A.50)

Clearly, we can define a rotation vector, , 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 -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,

 (A.51)

must be a vector as well. Also, if is interpreted as in Equation (A.49) then it follows that the equation of motion of a vector that precesses about the origin with some angular velocity is

 (A.52)

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