How to correctly identify r and r' in electromagnetics

[komender]

Homework Statement

This is more of a general problem that I noticed I constantly fail to get right, and seems like something quite basic in vector calculus but it mostly affects me in my electromagnetism class.
Whenever I try to work on a problem that involves figures and objects in space, I am struggling to get the position vectors right.
For example

It is basically the r' that gives me trouble almost every time.I can solve the rest of the question but for some reason I can't get the r'.

Homework Equations

The Attempt at a Solution

I can't quite explain my thought process here coherently but the way I see it, r' =phi*phi unit vector because it is the one that mostly defines the shape of the vector. I know this is a faulty logic and is incorrect, but I can't see it any other way (the right answer by the way is r'=a*rho unit vector

 

[mfb]

r is the point where you want to evaluate A?

r' is then along the loop. The loop is at $\rho=a$ and $z=0$, so two components in cylindrical coordinates are easy. Phi will vary along the integration path, and it is probably useful to use it as integration variable.