Angular Momentum Vector

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

where denotes the

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].

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