Solution

The basic steps used in the link to find the distance between P3 and P are:
1. Calculate the distance between P1 and P
2. Calculate the exact location of P
3. Calculate the distance between P and P3

Now, a little common sense suggests there are two too many steps here. But we can't compute the distance between P and P3 directly, because at this point the location of P is unknown.

Yet there's a very simple solution: we rotate P2 90 degrees around P1 to create P2'. By definition the distance between P1 and P' is the same as the distance between P3 and P. This has exactly the same form as the version in the link, but it is able to find the distance between P1 and P' in the first step, instead of the distance between P1 and P. Thus we've eliminated steps 2 and 3 entirely, and use exactly the same math as before:
u' = ( (x3 - x1)(x2' - x1) + (y3 - y1)(y2' - y1) ) / ( ||p2' - p1||^2 )

A visual proof:


While understanding the way the distance is calculated using the dot product requires knowledge of calculus/linear algebra, deriving this improved equation from the original in the link is just trivial high school geometry. That's why it's neat.