The two cross-products in Eq.(B.19) can be written out with the help of the vector analysis identityB.23
This (or a direct calculation) yields, starting with Eq.(B.19),
with
The vector angular momentum of a rigid body is obtained by summing the angular momentum of its constituent mass particles. Thus,
Since
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