# Radial component of velocity at extreme distances

Suppose I am given that a planet's position with respect to some star is of the form $$\textbf{r} = r\textbf{e}_{r}$$. Then of course $$\textbf{v} = \dot{r}\textbf{e}_{r} + r\dot{\theta}\textbf{e}_{\theta}$$.

Is it correct to say that, when the planet is at its closest and at its furthest from the star, the radial component of the velocity vector will always be zero, i.e. $$\dot{r}=0$$? In which case, the planet will still have some velocity purely in the $$\textbf{e}_{\theta}$$ direction?

Yes, the radial component of the velocity vector $$\dot{r}$$ is 0, since the distance between the planet and the origin is simply $$r$$. Assuming the planet is not on a purely radial trajectory, i.e. $$\dot{\theta}$$ is not zero at any point, the planet will also have velocity purely in the $$e_\theta$$ direction at this point.