I would like to ask for help with clarifying the following formula for calculation of relative position of a point on a line segment with respect to the line segment's length in two-dimensional Euclidean space:
$$t = \frac{(P-A)(B-A)}{\lVert B-A \rVert^2}$$
where P is a point of line segment AB and $t \in [0; 1]$.
This is a formula we have had in our OpenGL course but unfortunately our teacher skipped the explanation saing it is elementary.
I tried to derive the formula somehow from the standard Euclidean distance $t = \frac{\lVert P-A \rVert}{\lVert B-A \rVert}$ but I wasn't successful at all. The only other source I was able to find citing this formula was an old OpenGL on-line manual, however, it doesn't provide any explanation either. The formulas seem to be equivalent anyway.
Thank you.
$t$ is a scalar and can be defined as: $$t=\frac{||P-A||}{||B-A||}$$ Since $P$ is on the line segment $AB$, the vectors $P-A$ and $B-A$ would be in the same direction and $(P-A).(B-A)$ = $||P-A||||B-A||$. Using this condition and by multiplying numerator and denominator by $||B-A||$ in defintion of $t$, we arrive at the formula.