The unit vectors in plane polar co-ordinate system are orthogonal:

In a plane polar co-ordinate system, the radial unit vector is

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

and the angular unit vector is

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

[ To know the derivations of the above unit vectors CLICK HERE ]

Where, $\hat{i}$ and $\hat{j}$ are the unit vectors along the X-axis and Y-axis in two-dimensional cartesian co-ordinate system. Direction of the radial unit vector $\hat{r}$ is along the direction of the radial vector $\vec{r}$ and that of the angular unit vector $\hat{\theta}$ is along the direction of increasing $\theta$.

Two unit vectors are said to be orthogonal to each other when the angle between these two unit vectors is $90\circ$,

Now the dot product of the unit vectors $\hat{r}$ and $\hat{\theta}$ is given by,

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

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

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

[ since, $\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=1$ and $\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{i}=0$. ]

So the angle between two unit vectors $\hat{r}$ and $\hat{\theta}$ is $90^{\circ}$, i.e., the two unit vectors $\hat{r}$ & $\hat{\theta}$ are perpendicular to each other. This also indicates that the plane polar co-ordinate system is orthogonal.