Polar coordinates unit vectors proof

Question

Prove that the unit vectors in polar coordinates are related to those in rectangular coordinates by \begin{align*} \hat{r}&=\hat{x}\cos\phi+\hat{y}\sin\phi\\ \hat{\phi}&=-\hat{x}\sin\phi+\hat{y}\cos\phi. \end{align*} What are $\hat{x}$ and $\hat{y}$ in terms of $\hat{r}$ and $\hat{\phi}$?

Answer 1

$\hat{r}=\hat{x}\cos\phi+\hat{y}\sin\phi$ is a direct consequence of the unit circle definition of sine and cosine.

As $\hat{\phi}$ is the unit vector formed by rotating $\hat{r}$ through $90^\circ$,

\begin{align*} \hat{\phi}&=\hat{x}\cos(90^\circ+\phi)+\hat{y}\sin(90^\circ+\phi)\\ &=-\hat{x}\sin\phi+\hat{y}\cos\phi. \end{align*}

Answer 2

Another way to see the relation:

$\vec r=x\hat x+y\hat y$

$\hat r=\dfrac{\partial\vec r}{\partial r}/\left|\dfrac{\partial\vec r}{\partial r}\right|=\cos\phi\hat x+\sin\phi\hat y$$

$\hat\phi=\dfrac{\partial\vec r}{\partial\phi}/\left|\dfrac{\partial\vec r}{\partial\phi}\right|=(-r\sin\phi\hat x+r\cos\phi\hat y)/r=-\sin\phi\hat x+\cos\phi\hat y$