Like linear momentum, angular momentum is fundamentally a vector in
. The definition of the previous section suffices when the
direction does not change, in which case we can focus only on its
magnitude
.
More generally, let
denote the 3-space coordinates
of a point-mass
, and let
denote its velocity
in
. Then the instantaneous angular momentum vector
of the mass relative to the origin (not necessarily rotating about a
fixed axis) is given by
For the special case in which
is orthogonal to
, as in Fig.B.4, we have that
points, by the right-hand rule, in the direction of the angular
velocity vector
(up out of the page), which is
satisfying. Furthermore, its magnitude is given by
which agrees with the scalar case.
In the more general case of an arbitrary mass velocity vector
, we know from §B.4.12 that the magnitude of
equals the product of the distance from the axis
of rotation to the mass, i.e.,
, times the length of
the component of
that is orthogonal to
, i.e.,
, as needed.
It can be shown that vector angular momentum, as defined, is conserved.B.22 For example, in an orbit, such as that of the moon around the earth, or that of Halley's comet around the sun, the orbiting object speeds up as it comes closer to the object it is orbiting. (See Kepler's laws of planetary motion.) Similarly, a spinning ice-skater spins faster when pulling in arms to reduce the moment of inertia about the spin axis. The conservation of angular momentum can be shown to result from the principle of least action and the isotrophy of space [272, p. 18].