A doubt regarding this statement about motion in polar coordinate system There was this statement in my book that was not quite clear to me even after reading it after 20 times, the statement was:
There is a fundamental difference between polar and Cartesian base
vectors: the directions of $\vec{r}$ and $\vec{θ}$ vary with position, whereas $\vec{i}$ and $\vec{j}$ have
fixed directions
I can't seem to understand why can't we define a unit vector $\vec{r}$ amd $\vec{θ}$ in a predefined direction, because anyhow if we do so we can still take any linear combination of these unit vectors to get any other position, just like in the case of regular coordinate system where we predefined the unit vectors? What problem do we exactly face if we do the same thing in polar coordinates?
 A: 
I can't seem to understand why can't we define a unit vector $\vec{r}$ and $\theta$ in a predefined direction [...]

We can, but then they are not polar coordinates anymore. If you make the directions independent of position, all you will ever get is a skewed and/or rotated cartesian coordinate system. We often use polar coordinates to take advantage of symmetries in a problem. But to get a coordinate system where rotationally invariant problems become independent of one of the coordinates, we must use a coordinate system where $\vec{r}$ changes with $\theta$, because that's what the properties of the problem will also do.

In general, all unit vectors can depend on all coordinates. In the specific case of polar coordinates, both unit vectors only depend on the angle $\theta$.
If you imagine a circle (so constant radius $r$), the unit vectors are constantly changing as we move along that circle. $\vec{\theta}$ always points along the circle tangentially, and $\vec{r}$ always points radially outward. Both these directions depend on the angle $\theta$.
Now imagine moving along the radius. No directions change. Radially outward is the same direction, no matter how far from the origin you are. The tangent along the circle also doesn't change if you move further from or closer to the origin. So both unit vectors only depend on $\theta$.
A: I can't seem to understand why can't we define a unit vector $\vec r$ and $\vec \theta$  in a predefined direction,
You could but then how are the different from the vectors $\vec x$ and $\vec y$?
Polar coordinates are often used because it makes description of the motion of particles easier.  If that were not the case then it is unlikely that you would have heard of such a system.
