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.