What are $\hat{x}, \hat{y}, \hat{z}$ in terms of cylindrical unit vectors $\hat{r}, \hat{\theta}, \hat{z}$?

Question

I can find a lot of resources that show the inverse, (expressing cylindrical in terms of of Cartesian) but I just can't find what I want.

My wild guess is : $\hat{i} = -\sin{\theta} \;\hat{\theta} \\ \hat{j}=\cos{\theta}\;\hat{\theta} \\ \hat{z}=\hat{z}$

Answer 1

Cartesian $(x,y,z)$ to cylindrical $(\rho,\phi,z)$,

\begin{align} \hat{x} &= \cos\phi \hat{\rho} - \sin\phi \hat\phi \\ \hat{y} &= \sin\phi \hat{\rho} + \cos\phi \hat\phi \\ \hat{z} &= \hat z \end{align}

From Griffith's Introduction to Electyrodynamics, inside back cover.