Question

how cylindrical coordinate system are mutually perpendicular?

Answer 1

The cylindrical coordinate system is obtained as

$x=r\cos \phi , y=r\sin \phi , z=Z$

The ranges of $\hat{r}, \hat{\phi }, \hat{z}$ as

$0\leq r\leq \infty \\0\leq \phi \leq 2\pi \\ -\infty <z\leq \infty$

The Cartesian unit vector as of cylindrical coordinate system as

$\hat{r}=\cos \phi \hat{x}+\sin {\phi }\hat{y}$

$\hat{r}=-\sin \phi \hat{x}+\cos {\phi }\hat{y}$

$\hat{Z}=\hat{z}$

To verify that this unit vectors are mutually perpendicular.

$\hat{r}\cdot \hat{\phi }=0\\\hat{\phi}\cdot \hat{z }=0\\\hat{r}\cdot \hat{z }=0$

Thus the cylindrical coordinate system are mutually perpendicular.