14 deg, and it points along an axis midway between thexandy
axes.4.3.2 Angular momentum in three dimensions
The vector cross product
In order to expand our system of three-dimensional kinematics to
include dynamics, we will have to generalize equations likevt=ωr,
τ = rFsinθrF, andL = rpsinθrp, each of which involves three
quantities that we have either already defined as vectors or that we
want to redefine as vectors. Although the first one appears to differ
from the others in its form, it could just as well be rewritten as
vt=ωrsinθωr, sinceθωr= 90◦, and sinθωr= 1.
It thus appears that we have discovered something general about
the physically useful way to relate three vectors in a multiplicative
way: the magnitude of the result always seems to be proportional to
the product of the magnitudes of the two vectors being “multiplied,”
and also to the sine of the angle between them.
Is this pattern just an accident? Actually the sine factor has
a very important physical property: it goes to zero when the two
vectors are parallel. This is a Good Thing. The generalization of
angular momentum into a three-dimensional vector, for example, is
presumably going to describe not just the clockwise or counterclock-
wise nature of the motion but also from which direction we would
have to view the motion so that it was clockwise or counterclock-
wise. (A clock’s hands go counterclockwise as seen from behind the
clock, and don’t rotate at all as seen from above or to the side.) Now
suppose a particle is moving directly away from the origin, so that
itsrandpvectors are parallel. It is not going around the origin
from any point of view, so its angular momentum vector had better
be zero.
Thinking in a slightly more abstract way, we would expect the
angular momentum vector to point perpendicular to the plane of
motion, just as the angular velocity vector points perpendicular to
the plane of motion. The plane of motion is the plane containing
bothrandp, if we place the two vectors tail-to-tail. But ifrand
pare parallel and are placed tail-to-tail, then there are infinitely
many planes containing them both. To pick one of these planes in
preference to the others would violate the symmetry of space, since
they should all be equally good. Thus the zero-if-parallel property
is a necessary consequence of the underlying symmetry of the laws
of physics.
The following definition of a kind of vector multiplication is con-
sistent with everything we’ve seen so far, and on p. 1024 we’ll prove
that the definition is unique, i.e., if we believe in the symmetry of
space, it is essentially the only way of defining the multiplication of286 Chapter 4 Conservation of Angular Momentum