Families, images and bounces

You will write three functions. The first finds the multiplier $\lambda$ of the plane of a family that passes through a point. The second finds the image of a point in a plane. The third finds the shortest path from one point to another that touches a plane. A plane is stored as $(A, B, C, k)$ for $Ax + By + Cz = k$.

The idea. The plane $P_1 + \lambda P_2 = 0$ passes through a point when $\lambda = -\dfrac{P_1(\text{point})}{P_2(\text{point})}$. The image of $\vec{p}$ is $\vec{p} - 2\,\dfrac{\vec{p} \cdot \vec{n} - k}{\lvert \vec{n} \rvert^2}\,\vec{n}$. If $A$ and $B$ are on the same side of the plane, the shortest path has length $A'B$. If they are on opposite sides, it is $AB$.

Your tasks.

1. Finish family_lambda(plane1, plane2, point).
2. Finish image(point, plane). Return the image as a tuple.
3. Finish bounce_path(A, B, plane). Return the length of the shortest path from $A$ to the plane to $B$.