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

This (or a direct calculation) yields, starting with Eq.(B.19),

where

with , and , for . That is,

The matrix is the Cartesian representation of the

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}

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