# Components of a vector in polar coordinates

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.

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 $$3i\sin \left(\frac{π}{2}\right)$$.