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.