3D angular velocity

Now consider the rotation of a 3D rigid body. Recall from Section 3.3 that Euler's rotation theorem implies that every 3D rotation can be described as a rotation $\theta$ about an axis $v = (v_1, v_2, v_3)$ though the origin. As the orientation of the body changes over a short period of time $\Delta t$, imagine the axis that corresponds to the change in rotation. In the case of the merry-go-round, the axis would be $v = (0,1,0)$. More generally, $v$ could be any unit vector.

The 3D angular velocity is therefore expressed as a 3D vector:

$$(\omega_x,\omega_y,\omega_z),$$ (8.15)

which can be imagined as taking the original $\omega$ from the 2D case and multiplying it by the vector $v$. This weights the components according to the coordinate axes. Thus, the components could be considered as $\omega_x = \omega v_1$, $\omega_y = \omega v_2$, and $\omega_z = \omega v_3$. The $\omega_x$, $\omega_y$, and $\omega_z$ components also correspond to the rotation rate in terms of pitch, roll, and yaw, respectively. We avoided these representations in Section 3.3 due to noncommutativity and kinematic singularities; however, it turns out that for velocities these problems do not exist [308]. Thus, we can avoid quaternions at this stage.