Editing Angular velocity (section)

=== Components from the basis vectors of a body-fixed frame ===

Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 </math> fixed to the body and with their common origin at O. The spin angular velocity vector of both frame and body about O is then
: <math>\boldsymbol\omega = \left(\dot \mathbf{e}_1\cdot\mathbf{e}_2\right) \mathbf{e}_3 + \left(\dot \mathbf{e}_2\cdot\mathbf{e}_3\right) \mathbf{e}_1 + \left(\dot \mathbf{e}_3\cdot\mathbf{e}_1\right) \mathbf{e}_2,
</math>
where <math> \dot \mathbf{e}_i= \frac{d  \mathbf{e}_i}{dt} </math> is the time rate of change of the frame vector <math> \mathbf{e}_i, i=1,2,3,</math>  due to the rotation.

This formula is incompatible with the expression for ''orbital'' angular velocity

: <math>\boldsymbol\omega
=\frac{\boldsymbol{r}\times\boldsymbol{v}}{r^2},</math>

as that formula defines angular velocity for a ''single point'' about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a ''single'' <math> \boldsymbol\omega</math> has to account for the motion of ''all'' particles in the body.