Sorry if the title is a bit ambiguous, let me clarify things with a picture first, things written in black are known.
This is the way I would approach the problem; my trig skills are highly lacking so please correct the eventual mistakes or redundant calculations.
In order to figure out the angle between the two vectors I use this formula:
$ \theta = \arctan(a/b) $
Now to get the $x'$ coordinate
$x' = x + r \cos(\theta) $
and for the $y'$
$y' = y + r \sin(\theta)$
Is this correct? I feel like there's something missing or a simpler way to do it. Thanks.
That works, although you’re going to run into trouble if $b=0$. There’s no need to compute $\theta$ explicitly, though. Observe that $\cos\theta={b\over\sqrt{a^2+b^2}}$ and $\sin\theta={a\over\sqrt{a^2+b^2}}$. Substitute these into the formulas that you have for $x'$ and $y'$. Essentially, what that does is scale the vector $(b,a)$ so that its length is $r$ and then adds it to $c$.