Tangential Velocity as a Cross Product

Referring again to Fig.B.4, we can write the tangential velocity vector $\underline{v}$ as a vector cross product of the angular-velocity vector $\underline {\omega }$ (§B.4.11) and the position vector $\underline {x}$ :

$$\underline{v}\eqsp \underline{\omega}\times \underline{x} \protect$$ (B.17)

To see this, let's first check its direction and then its magnitude. By the right-hand rule, $\underline {\omega }$ points up out of the page in Fig.B.4. Crossing that with $\underline {x}$ , again by the right-hand rule, produces a tangential velocity vector $\underline{v}$ pointing as shown in the figure. So, the direction is correct. Now, the magnitude: Since $\underline {\omega }$ and $\underline {x}$ are mutually orthogonal, the angle between them is $\pi /2$ , so that, by Eq.(B.16),

$$\left\Vert\,\underline{\omega}\times \underline{x}\,\right\Vert \eqsp \left\Vert\,\underline{\omega}\,\right\Vert\cdot\left\Vert\,\underline{x}\,\right\Vert \eqsp \omega R \eqsp v$$

as desired.