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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 righthand 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.105 shows the effect of applying two successive rotations,
one about , and the other about the , to a standard sixsided die. In the
lefthand case, the rotation is applied before the rotation, and vice
versa in the righthand 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 noncommuting algebra cannot be
represented by vectors. So, although rotations have a welldefined magnitude and
direction, they are not vector quantities.
Figure A.105:
Effect of successive rotations about perpendicular axes on a sixsided die.

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 the by
.
According to Equations (A.1280)(A.1282), we have

(1305) 
where use has been made of the small angle approximations
and
. The above equation can easily be generalized to allow
small rotations about and by
and
,
respectively. We find that

(1306) 
where

(1307) 
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 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,

(1308) 
must be a vector as well. Also, if is interpreted as
in Equation (A.1306) then it follows that the equation of motion of a vector
which precesses about the origin with some angular velocity is

(1309) 
Next: Scalar Triple Product
Up: Vector Algebra and Vector
Previous: Vector Product
Richard Fitzpatrick
20110331