In a two-dimensional plane polar co-ordinate system, there are two unit vectors \cdots
- The radial unit vector \hat{r} . The direction of this unit vector is along the radius vector \vec{r}
- The unit vector \hat{\theta} along the direction of increasing \theta .
Similarly, in case of planer motion in a two-dimensional Cartesian co-ordinate system, there are also two unit vectors,
- The unit vector \hat{i} along the direction of the X-axis.
- The unit vector \hat{j} along the direction of the Y-axis
Relations:
- The unit vector \hat{r} in term of the unit vectors \hat{i} and \hat{j} :
Let, In two dimensional cartesian co-ordinate system, (x,y) be the position vector of the point A
And, in plane polar co-ordinate system, ( \hat{r} , \theta ) be the position vector of the same point A , as shown in the figure 1.1.
Here, \vec{OA}=\vec{r} .
So the unit vector \hat{r} is along the direction of the vector \vec{r} , i.e., \vec{OA} . The unit vector \hat{\theta} is perpendicular to the vec{OA} .
The unit vector \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}} , where, |\vec{r}| is the modulus of the vector \vec{r} .
Now, \vec{r}=x\hat{i}+y\hat{j}
Again, x=r\cos\theta & y=r\sin\theta
therefore, \vec{r}=r\cos\theta\hat{i}+r\sin\theta\hat{j}
so the unit vector, \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}=\frac{ r\cos\theta\hat{i}+r\sin\theta\hat{j}}{r}}
or, \hat{r}=\cos\theta\hat{i}+\sin\theta\hat{j} .

- The unit vector \hat{\theta} in term of the unit vectors \hat{i} and \hat{j} :
Let us consider a point A having polar co-ordinate (r,\theta) & another point B having polar co-ordinate (r,\theta+d\theta) , as shown in the adjoining figure 1.2.
\vec{OA}=r\cos\theta\hat{i}+r\sin\theta\hat{j}
and, \vec{OB}=r\cos(\theta+d\theta)\hat{}+r\sin(\theta+d\theta)\hat{j}
Again, \vec{AB}=\vec{OA}-\vec{OB} .
or, \vec{AB}= r\cos\theta\hat{i}+r\sin\theta\hat{j}- r\cos(\theta+d\theta)\hat{}+r\sin(\theta+d\theta)\hat{j}
or, \vec{AB}=r[\cos(\theta+d\theta)-\cos\theta]\hat{i}+r[\sin(\theta+d\theta)-\sin\theta]\hat{j}
or \vec{AB}=-r\ \sin\theta\ d\theta\hat{i}+r\ \cos\theta\ d\theta\hat{j}
Where, \cos(\theta+d\theta)-\cos\theta=\cos\theta\ \cos{d\theta}-\sin\theta\ \sin{d\theta}-\cos\theta
or, \cos(\theta+d\theta)-\cos\theta=-\sin\theta\ d\theta
[ Since, d\theta is very small. So \cos{d\theta}=1 and \sin{d\theta}=d\theta ]
Similarly, \sin(\theta+d\theta)-\sin\theta=\sin\theta\ \cos{d\theta}-\cos\theta\ \sin{d\theta}-\sin\theta
or, \sin(\theta+d\theta)-\sin\theta =\cos\theta\ d\theta
The direction of the vector \vec{AB} is along the direction of the increasing \theta .
The magnitude of the vector \vec{AB} is,
|\vec{AB}|=r\ d\theta\sqrt{({\sin}^2\theta+{\cos}^2\theta)}=r\ d\theta
The unit vector
\displaystyle{\hat{\theta}=\frac{\vec{AB}}{|\vec{AB}|}}
or, \displaystyle{\hat{\theta}=\frac{-r\ \sin\theta\ d\theta\hat{i}+r\ \cos\theta\ d\theta\hat{j}}{r\ d\theta}}
or, \hat{\theta}=-\sin\theta\hat{i}+\cos\theta\hat{j}

The polar unit vectors vary from point to point, they are not fixed like \hat{i} and \hat{j} . When a point moves, the polar unit vectors become a function of time.