### 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

*vector cross product*, discussed in §B.4.12 above. The identity was discussed at Eq.(B.17).

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

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
[270, p. 18].

#### Angular Momentum Vector in Matrix Form

The two cross-products in Eq.(B.19) can be written out with the help
of the vector analysis identity^{B.23}

where

The matrix is the Cartesian representation of the

*mass moment of inertia tensor*, which will be explored further in §B.4.15 below.

The vector angular momentum of a rigid body is obtained by summing the angular momentum of its constituent mass particles. Thus,

In summary, the angular momentum vector is given by the mass moment of inertia tensor times the angular-velocity vector representing the axis of rotation.

Note that the angular momentum vector
does *not* in general
point in the same direction as the angular-velocity vector
. We
saw above that it does in the special case of a point mass traveling
orthogonal to its position vector. In general,
and
point
in the same direction whenever
is an *eigenvector* of
, as will be discussed further below (§B.4.16). In this
case, the rigid body is said to be *dynamically balanced*.^{B.24}

**Next Section:**

Mass Moment of Inertia Tensor

**Previous Section:**

Angular Momentum