Let us consider a point P in cylindrical polar co-ordinate system, having cylindrical co-ordinates $(\rho,\ \phi,\ z)$. The cartesian co-ordinates of the point P is $(x,\ y,\ z)$. Let’s draw a vertical line PQ on the X-Y plane, which is parallel to the z-axis, as shown in Fig.1.

OQ is represented by $\rho$, which makes an angle $\phi$ with the X-axis.

From Fig. 1 we can write,

$x=\rho\ \cos\phi$

$y=\rho\ \sin\phi$

and $z=z$

If $\vec{r}$ be the position vector with respect to the origin O, then we can write $\vec{r}=x\ \hat{i}+y\ \hat{j}+z\ \hat{k}\tag{1}$

where, $\hat{i}$, $\hat{j}$ and $\hat{k}$ is the unit vector along the X-axis, Y-axis and Z-axis respectively.

Now, equation (1) can be written as,

$\vec{r}=\rho\ \cos\phi\ \hat{i}+\rho\ \sin\phi\ \hat{j}+z\ \hat{k}\tag{2}$

Let, $\hat{\rho}$ be the unit vector along the direction of increasing $\rho$, then we can write,  

$\displaystyle{\hat{\rho}=\frac{\frac{\delta{\vec{r}}}{\delta{\rho}}}{|\frac{\delta{\vec{r}}}{\delta{\rho}}|}}$

Now, $\frac{\delta{\vec{r}}}{\delta{\rho}}=\cos\phi\ \hat{i}+\sin\phi\ \hat{j}$

Therefore, $\hat{\rho}=\cos\phi\ \hat{i}+ \sin\phi\ \hat{j}\tag{i}$

Let, $\hat{\phi}$ be the unit vector along the direction of increasing $\phi$ then we can write,

$\displaystyle{\hat{\phi}=\frac{\frac{\delta{\vec{r}}}{\delta{\phi}}}{|\frac{\delta{\vec{r}}}{\delta{\phi}}|}}$

Now, $\frac{\delta{\vec{r}}}{\delta{\phi}}=-\rho\ \sin\phi\ \hat{i}+\rho\ \cos\phi\ \hat{j}$

and $|\frac{\delta{\vec{r}}}{\delta{\phi}}|= \sqrt{{\rho}^2\ \sin^2\phi+{\rho}^2\ \cos^2\phi}=\rho$

Therefore, $\hat{\phi}=\displaystyle{\frac{\frac{\delta{\vec{r}}}{\delta{\phi}}}{|\frac{\delta{\vec{r}}}{\delta{\phi}}|}}=-\sin\phi\ \hat{i}+\cos\phi\ \hat{j}\tag{ii}$

Let, $\hat{z}$ be the unit vector along the direction of increasing $z$, then we can write,

$\displaystyle{\hat{z}=\frac{\frac{\delta{\vec{r}}}{\delta{z}}}{|\frac{\delta{\vec{r}}}{\delta{z}}|}}$

Now, $\displaystyle{\frac{\delta{\vec{r}}}{\delta{z}}=\hat{k}}$

Therefore, $\hat{z}=\hat{k}\tag{iii}$

So, the equations (i), (ii) and (iii) represent the unit vectrors in cylindrical polar co-ordinate system in terms of the unit vectors in spherical polar co-ordinate system.