What are the coordinates of the foot of the perpendicular drawn from the point (4, 1) to the line $x+y=3$?

The given line has slope −1, so the perpendicular through (4, 1) has slope 1: $y-1=x-4$, that is $y=x-3$. Substituting into $x+y=3$ gives $x+x-3=3$, so $x=3$ and $y=0$. The foot is (3, 0), and it does lie on the line since 3 + 0 = 3.