If I know the two Vectors $v_1$ and $v_2$, which describe points in a 2D space, and I also know that a vector $v_3$ is on the line segment between $v_1$ and $v_2$, how can I get the $x$ and $y$ coordinates of $v_3$ if the only thing I know about $v_3$ is the hypotenuse between $v_3$ and $v_1$. Sadly I dont know much about interpolation, the stuff I read on the internet is always about "if you know what $v_3\cdot x$ is you can calculate $v_3\cdot y$" but that is not what I want and I can't think of a way to transform this formula so that I can use it.
I though it should be able to do this by using the intersection-theory I had at school long ago, but I'm also unable to find a solution for my problem using this approach.
I would have liked to post an image for illustration purposes, but I don't have enough rep.
Let $|\vec v_1-\vec v_2|=d$ and $|\vec v_1-\vec v_3|=a$. If I well understand your question you know $ \vec v_1=(x_1,y_1)^T$,$\vec v_2=(x_2,y_2)^T$, $d$ and $a$ and you want the coordinates of $\vec v_3$ on the segment $\vec v_1-\vec v_2$.
Hint:
Usethe fact that a vector on the segment is given by: $$ (x_3,y_3)^T=(x_1,y_1)^T+t(x_2-x_1,y_2-y_1)^T \quad \mbox{for}\quad t\in[0,1] $$ and: $|\vec v_3-\vec v_1|=t|\vec v_2-\vec v_1|$
for $t=a/d$ you have the result.