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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.

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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 $3i\sin \left(\frac{π}{2}\right)$.

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  • $\begingroup$ Thanks for the answer, so how did you use the relation between the basis vectors that the exercise asks to find? $\endgroup$
    – Slayer147
    May 3 '19 at 11:24

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