I wanted to know how can I align my A vertex to my B vertex while keeping a straight and vertex normal's based line. Thanks !
First, the line you're interested in is not the normal of anything. "Normal" is a term with a very particular meaning in 3D graphics. If you're not yet familiar with its usage in 3D, then try to use other words, or else people will give you wrong answers-- "wrong" because they won't do what you wanted them to do. (A has a vertex normal. But you do not want to move A along its vertex normal.)
Okay, so we want to move A to the end of the dotted line-- not the B end (I will call the desired point "C".) That dotted line is perpendicular to the edge containing A that does not also contain B (I will call that edge AD.) How do we do that?
In edit mode, select edge AD. "Create custom orientation" operation (hotkey ctrl alt space for me.) We see our transform orientation change to "Edge".
Select B. Snap cursor to selected (hotkey shift-s menu for me.) Change transform pivot point, on the top of your 3D viewport, to cusor (hotkey . for me.)
"S y 0 [enter]" to scale A, along the edge AD, along our custom orientation's Y axis, to a position as close to B as possible.