# 7.1: Linear and Angular Velocity

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We related the linear and angular velocities of a rotating object in two dimensions in Section 5.1. There, we also already stated the relation between the linear velocity vector and rotation vector in three dimensions (Equation 5.1.5):

$\boldsymbol{v}=\boldsymbol{\omega} \times \boldsymbol{r}$

It is not hard to see that this expression indeed simplifies to the scalar relationship $$v = \omega r$$ for rotations in a plane, with the right sign for the linear velocity. That’s hardly a proof though, so let’s put this on some more solid footing. Suppose $$\boldsymbol{r}$$ makes an angle $$\phi$$ with $$\boldsymbol{\omega}$$. Suppose also that it changes by $$\mathrm{d}\boldsymbol{r}$$ in a time interval $$\mathrm{d}t$$, then if we have pure rotation, $$\mathrm{d}\boldsymbol{r}$$ is perpendicular to both $$\boldsymbol{r}$$ and $$\boldsymbol{\omega}$$, and its magnitude is given by $$|d \boldsymbol{r}| = \omega r \sin \phi \mathrm{d} t=|\boldsymbol{\omega} \times \boldsymbol{r}| \mathrm{d} t$$, where $$\omega$$ and $$r$$ are the lengths of their respective vectors. Finally, as seen from the top (i.e., looking down the vector $$\boldsymbol{\omega}$$), the rotation should be counter-clockwise (by definition of the direction of $$\boldsymbol{\omega}$$), which corresponds with the direction of $$\boldsymbol{\omega} \times \boldsymbol{r}$$. We thus find that both the magnitude and direction of $$\boldsymbol{v}=\mathrm{d} \boldsymbol{r} / \mathrm{d} t$$ indeed equal $$\boldsymbol{\omega} \times \boldsymbol{r}$$, and Equation 5.1.5 holds.

This page titled 7.1: Linear and Angular Velocity is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Timon Idema (TU Delft Open) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.