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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?

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closed as off-topic by John Rennie, ACuriousMind May 18 '17 at 10:36

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – John Rennie, ACuriousMind
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual question over those just asking for a specific computation. $\endgroup$ – ACuriousMind May 18 '17 at 10:36
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One way would be to use the definition of the dot product for vectors

$$ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos \theta, $$ hence $$ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}.$$

I've done the calculation quickly and you will definitely end up with a constant angle :)

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Consider how you would compute the angle between two vectors in Cartesian coordinates, we would take $cos^{-1}$ of $\frac{\vec{v} \cdot \vec{a}}{|\vec{v}||\vec{a}|}$. Since $\hat{r}$ and $\hat{\theta}$ are mutually perpendicular, just like $\hat{i}$ and $\hat{j}$, we can think of the $\hat{r}$ and $\hat{\theta}$ components as a rotation of the $\hat{i}$ and $\hat{j}$ coordinates. As angles between vectors are preserved under rotation, this analysis should give the correct answer for the angle between the two vectors.

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