Unit vectors in spherical polar co-ordinate system are mutually perpendicular to each other:

Two unit vectors are said to be orthogonal to each other when the angles between these two unit vectors are $90^{\circ}$, i.e., they are perpendicular to each other.

In order to prove the unit vectors ($\hat{r}$, $\hat{\theta}$, $\hat{\phi}$) in spherical polar co-ordinate system are mutually perpendicular to each other, we have to prove that the angles between $\hat{r}$ & $\hat{\theta}$, $\hat{\theta}$ & $\hat{\phi}$, $\hat{r}$ & $\hat{\phi}$ are $90^{\circ}$ each.

Let us consider a point A having spherical polar co-ordinate ($r,\theta,\phi$) as shown in figure 1.

The unit vectors in three-dimensional polar co-ordinate system are $\hat{r}$, $\hat{\theta}$ , $\hat{\phi}$.

So we can write,

$\hat{r}=\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k}$

$\hat{\theta}=\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k}$

$\hat{\phi}=-\sin\phi\ \hat{i}+\cos\phi\ \hat{j}$

where, $\hat{i}.\hat{j},\hat{k}$ are the unit vectros in the three-dimensional cartesian co-ordinate system along X-axis, Y-axis and Z-axis respectively.

[ To know the derivation of the above unit vectors, CLICK HERE ]

Angle between $\hat{r}$ & $\hat{\theta}$:

The scaler product of $\hat{r}$ & $\hat{\theta}$ is given by,

$\hat{r}\cdot\hat{\theta}=(\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k})\cdot(\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k})$

or, $\hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta\ {\cos}^2\phi+\sin\theta\ \cos\theta\ {\sin}^2\phi-\cos\theta\ \sin\theta$

or, $\hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta({\cos}^2\phi+{\sin}^2\phi)-\sin\theta\ \cos\theta$

or, $\hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta-\sin\theta\ \cos\theta$

or, $\hat{r}\cdot\hat{\theta}=0$

[where, $\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{i}\cdot\hat{k}=0$ and $\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1$]

So the angle between $\hat{r}$ and $\hat{\theta}$ is $90^{\circ}$, i.e., the unit vectors $\hat{r}$ and $\hat{\theta}$ are perpendicular to each other.

Angle between $\hat{\theta}$ & $\hat{\phi}$:

The scalar product of the unit vectors $\hat{\theta}$ and $\hat{\phi}$ is given bellow,

$\hat{\theta}\cdot\hat{\phi}=(\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k})\cdot(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})$

or, $\hat{\theta}\cdot\hat{\phi}=-\cos\theta\ \sin\phi\ \cos\phi+\cos\theta\ \sin\phi\ \cos\phi$

or, $\hat{\theta}\cdot\hat{\phi}=0$

So the angle between the unit vectors $\hat{\theta}$ & $\hat{\phi}$ is  $90^{\circ}$, i.e., the unit vectors $\hat{\theta}$ & $\hat{\phi}$ are perpendicular to each other.

Angle between $\hat{r}$ & $\hat{\phi}$:

The scalar product of the unit vectors $\hat{r}$ and $\hat{\phi}$ is given bellow,

$\hat{r}\cdot\hat{\phi}=(\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k})\cdot(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})$

or, $\hat{r}\cdot\hat{\phi}=-\sin\theta\ \sin\phi\ \cos\phi+\sin\theta\ \sin\phi\ \cos\phi$

or, $\hat{r}\cdot\hat{\phi}=0$

So the angle between the unit vectors $\hat{r}$ & $\hat{\phi}$ is $90^{\circ}$, i.e., the unit vectors $\hat{r}$ & $\hat{\phi}$ are perpendicular to each other.

Therefore the unit vectors $\hat{r}$, $\hat{\theta}$ and $\hat{\phi}$ in the spherical polar co-ordinate system are mutually perpendicular to each other.