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 , and , for . That is,
The vector angular momentum of a rigid body is obtained by summing the angular momentum of its constituent mass particles. Thus,
Since factors out of the sum, we see that the mass moment of inertia tensor for a rigid body is given by the sum of the mass moment of inertia tensors for each of its component mass particles.
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