I was studying a Spherical Coordinate System. And I kinda stuck in process where it's coordinate is represent in term of vector notation . Any point $P$ in standard vector space can be represented by three unit vectors $\mathbf i,\ \mathbf j$ and $\mathbf k$.
But in case of Spherical Coord. Sys. there are different basis vector $\theta,\ r$ and $ρ$. As we know magnitude of $\mathbf{i, \ j, \ k}$ are unit or $1$. How about magnitude $\rho$. Is it unit? But $ρ$ is an angle. How it can be unit (as unit is for distance)? Is it related to polar coordinate system?
Can anyone please explain me how does it all happen and how does same point can be represented by both Cartesian and spherical vector coordinate system?
What do you think the basis vectors might be in a system of spherical co-ordinates ? The vector from the origin to $r=1, \theta=0, \phi=0$ is simple enough. But what about the vector from the origin to $r=0, \theta=1, \phi=0$ ? Or to $r=0, \theta=0, \phi=1$ ?
The problem with spherical co-ordinates (or any system of non-Cartesian co-ordinates) is that the direction from a point $P$ along which one co-ordinate changes and the other two remain constant depends on the location of $P$. In Cartesian co-ordinates the directions and lengths of the basis vectors are independent of location. But you are trying to generalise this property of Cartesian co-ordinates to other co-ordinate systems where it just does not hold.