Foot of the perpendicular by computer

You will write three functions about a point and a line. The first finds the foot of the perpendicular, the second finds the distance, and the third finds the two points of the line that are at a given distance from the point.

The idea. For the line through $A$ with direction $\vec{b}$, the foot of the perpendicular from $P$ is $F = A + t\vec{b}$ with $t = \dfrac{(P - A) \cdot \vec{b}}{\lvert \vec{b} \rvert^2}$. The distance is $PF$. A point $Q$ of the line at distance $s$ from $P$ lies $\sqrt{s^2 - PF^2}$ from the foot, along the line.

Your tasks.

1. Finish foot(P, A, b). Return the foot as a tuple of three numbers.
2. Finish distance_to_line(P, A, b). Return the length of $PF$.
3. Finish points_at(P, A, b, s). Return the two points $F \pm \dfrac{\sqrt{s^2 - PF^2}}{\lvert \vec{b} \rvert}\vec{b}$, with the point of the larger parameter first.