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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 righthand 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 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. Clearly, 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 9:

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
20060202