Converting vector in cartesian to cylindrical coordinates

Question

This seems like a trivial question, and I'm just not sure if I'm doing it right.
I have vector in cartesian coordinate system: $\vec{N} =y\vec{a_x} −2x\vec{a_y} + y\vec{a_z}$. And I need to represent it in cylindrical coord.
Relevant equations: $$A_\rho=A_xcos\phi+A_ysin\phi$$ $$A_\phi=−A_xsin\phi+A_ycos\phi$$ $$A_z=A_z$$
What is cofusing me is this: The formula for $\phi$ is $\phi=arctan(\frac{y}{x})$ . Are those $x$ and $y$ in fact $a_x$ and $a_y$? If so, then for my problem, wouldn't it be $\phi=arctan(\frac{-2x}{y})$? And do I need to change the unit vectors too?