FIGURE 12.44 shows a rotating rigid body. We can define an angular velocity vector $\vec{\omega }$ as follows:

• The magnitude of $\vec{\omega }$ is the object’s angular speed $\text{|}\omega \text{|}$.

• $\vec{\omega }$ points along the axis of rotation in the direction given by the right-hand rule illustrated in FIGURE 12.44.

FIGURE 12.44 The angular velocity vector $\vec{\omega }$ is found using the right-hand rule.

If the object rotates in the xy-plane, the vector $\vec{\omega }$ points along the z-axis. The scalar angular velocity $\omega =v_t\text{/}r$ that we’ve been using is now seen to be ${\omega }_z$, the z-component of the vector $\vec{\omega }$. You should convince yourself that the sign convention for $\omega$ (positive for ccw rotation, negative for cw rotation) is equivalent to having the vector $\vec{\omega }$ pointing in the positive z-direction or the negative z-direction.