Angular Velocity Vector

When working with rotations, it is convenient to define the
*angular-velocity vector* as a vector
pointing
along the *axis of rotation*. There are two directions we could
choose from, so we pick the one corresponding to the *right-hand
rule*, *i.e.*, when the fingers of the right hand curl in the direction
of the rotation, the thumb points in the direction of the angular
velocity vector.^{B.18} The
*length*
should obviously equal the angular
velocity
. It is convenient also to work with a unit-length
variant
.

As introduced in Eq.
(B.8) above, the mass moment of inertia is
given by
where
is the distance from the (instantaneous)
axis of rotation to the mass
located at
. In
terms of the angular-velocity vector
, we can write this as
(see Fig.B.6)

where

denotes the orthogonal projection of onto (or ) [454]. Thus, we can project the mass position onto the angular-velocity vector and subtract to get the component of that is orthogonal to , and the length of that difference vector is the distance to the rotation axis , as shown in Fig.B.6.

Using the *vector cross product* (defined in the next section),
we will show (in §B.4.17) that
can be written more succinctly as

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University