### Cylindrical and spherical coordinates

We know that
$\hat{\rho}=\cos\varphi \hat{x}+ \sin\varphi \hat{y} \hat{-\varphi}=-\sin\varphi \hat{x}+ \cos\varphi \hat{y}$
Let, for example, $\hat{\rho}=a_1\hat{r}+a_2\hat{\theta}+a_3\hat{\varphi}$. Like it was in the previous problem,
$a_1=\hat{\rho}\cdot\hat{r}= \sin\theta\\ a_2=\hat{\rho}\cdot\hat{\theta}=\cos\theta\\ a_3=\hat{\rho}\cdot\hat{\varphi}=0$
and similarly for $\hat{\varphi}$ and $\hat{z}$.