Is rotation a vector?

The rotation ``vector''
now has a well-defined magnitude and
direction. But, is this quantity really a vector?
This may seem like a strange question to ask, but it turns out that not all
quantities which have well-defined magnitudes and directions are necessarily
vectors. Let us review some properties of vectors. If and
are two general vectors, then it is certainly the case that

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There is a direct analogy between rotation and motion over the Earth's surface. After
all, the motion of a pointer along the Earth's equator from longitude W to
longitude W could just as well be achieved by keeping the pointer fixed and
rotating the Earth through about a North-South axis. The non-commutative nature
of rotation ``vectors'' is a direct consequence of the non-planar (*i.e.*, curved)
nature of the Earth's surface.
For instance, suppose we start off at (N, W), which is just off the Atlantic
coast of equatorial Africa, and rotate northwards and then eastwards.
We end up at (N, E), which is in the middle of the Indian Ocean. However,
if we start at the same point, and rotate eastwards and then northwards,
we end up at the North pole. Hence, large rotations over the Earth's surface do
not commute.
Let us now repeat this experiment on a far smaller
scale. Suppose that we walk 10m northwards and then 10m eastwards.
Next, suppose that--starting from
the same initial position--we walk 10m eastwards and then 10m northwards. In this case, few people
would need much convincing that the two end points are essentially identical. The
crucial point
is that for sufficiently small displacements the Earth's surface is approximately planar, and
vector displacements on a plane surface commute with one another. This observation immediately
suggests that rotation ``vectors'' which correspond to rotations through *small angles*
must also commute with
one another. In other words, although the quantity
, defined above, is not a true
vector, the infinitesimal quantity
, which is defined in a similar manner but
corresponds to a rotation through an infinitesimal angle , is a perfectly good
vector.

We have just established that it is possible to define a true vector
which
describes a rotation through a *small* angle about a fixed axis. But, how is this
definition
useful? Well, suppose that vector
describes the small rotation that a given
object executes in the infinitesimal time interval between and . We can
then define the quantity

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Suppose, for example, that a rigid body rotates at constant angular velocity
.
Let us now combine this motion with rotation about a *different axis* at constant
angular velocity
. What is the subsequent motion of the body? Since we know
that angular velocity is a vector, we can be certain that the combined motion simply
corresponds to rotation about a third axis at constant angular velocity

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