Components of a vector in polar coordinates

Question

The problem is the following:

A vector in the $xy$ plane starts at the point $(0,-2)$ and ends (where the tip is) at position $(0,1)$. Write its components in polar coordinates.

The problem suggests to start writing the relation between the basis vectors

$\partial_x = \cos\theta \ \partial_r -\dfrac{\sin\theta}{r} \ \partial_\theta$ and $\partial_y=\sin\theta \ \partial_r + \dfrac{\cos\theta}{r} \ \partial_\theta$

I got to the point above, but doesn't know how to proceed from here.

Note: I am starting to learn differential geometry so any clues on how to proceed will be helpful.

Answer 1

The vector starts at $(0,-2)$ and ends at $(0,1)$, therefore $\vec{v}=[0,3]$.

Notice that the horizontal component is $0$ and the vertical component is $3$, so the magnitude is simply $3$.

The angle (argument) is $\displaystyle \frac{π}{2}$ because the vector points upward.

Written in polar coordinates, it could be written as $\displaystyle (r, \theta)=\left(3,\frac{π}{2}\right)$, or $\large \displaystyle 3e^{\frac{π}{2}}$, or $3i\sin \left(\frac{π}{2}\right)$.