The problem was to find the scale factors $$h_R,h_\theta,h_z$$. For the cylindrical polar coordinate system, and hence obtain expressions for the unit vectors $$ \bar{e_R},\bar{e_\theta},\bar{e_z}$$ and then verify that this is an orthogonal coordinate system.
My attempt: $x=R\cos(\theta)$, $y=R\sin(\theta)$,$z=z$ so we have that $$h_r=1,$$ $$h_\theta=R,$$ $$h_z=1.$$ My problem lies here, i dont know how to calculate the unit vectors i was thinking that $$\vec{e_R}=\frac{\frac{\partial \vec{r}}{\partial R}} {\left|\frac{\partial \vec{r}}{\partial R}\right|} $$ where $\vec{r}=xi+yj+zk$ but i'm unsure if this is the correct approach, and then after finding them i assume i dot them with one another to show that the coordinate system is orthogonal would they need to be equal to $0$ when i do this to show that the system is orthogonal. Any help would be appreciated, thanks.