In polar co-ordinates we have
$\vec{r} = r\hat{r}$ and $\vec{v} = \dot{r}\hat{r} + r\dot{\theta}\hat{\theta}$ and $\vec{a} = (\ddot{r}-r\dot{\theta}^2)\hat{r} + 2\dot{r}\dot{\theta}\hat{\theta}$
Now suppose we had $r(t) = be^{wt}$ and $\theta(t) = wt$. Then $\vec{v} = wbe^{wt}(\hat{r} + \hat{\theta})$ and $\vec{a} = w^2be^{wt}\hat{r} + 2w^2be^{wt}\hat{\theta}$
How could I show that the angle between $\vec{v}$ and $\vec{a}$ is constant?