If a point-mass is located at and is rotating about an axis-of-rotation with angular velocity , then the distance from the rotation axis to the mass is , or, in terms of the vector cross product, . The tangential velocity of the mass is then , so that the kinetic energy can be expressed as (cf. Eq. (B.23))
In a collection of masses having velocities , we of course sum the individual kinetic energies to get the total kinetic energy.
Finally, we may also write the rotational kinetic energy as half the inner product of the angular-velocity vector and the angular-momentum vector:B.27
where the second form (introduced above in Eq. (B.7)) derives from the vector-dot-product form by using Eq. (B.20) and Eq. (B.22) to establish that .