Find point along vector given a distance along axis

Question

So I have to find a point along a vector where the x component has to be a certain length. I made a drawing with my paint skills to explain this problem.

Image

So in the drawing, the distance between the x component of point B and A is like 1000(xB - xA = 1000). I need to find a point( along the vector $\vec {AB}$) xA' where the distance between the xA' and xA is let's say 100. I will then have the vector $\vec {AA'}$ and the x component between xA' and xA will be 100.

I know I can do B - A = $\vec {AB}$,

Normalize it : $\widehat{AB}$

Then I can add different lengths to xA until I find the length where :

Then $\widehat{AB}$ $\cdot$ length = $\vec {AA'}$

But is there another way algebraically?

Thank you!