I am currently working on some homework which relates to perspective projects. I've been able to complete some of the questions but not others. I'd like to be pointed in the correct direction and have my reasoning corrected, if needed.

Say that we've given 4 points, p_0, p_1, p_2 and p_3, retrieved from a perspective projection of a 3D rectangle, where the sides aren't parallel. If the focal point is at the origin and the camera is looking down the negative z-axis:

My idea is that create two lines from 2 points each and find their intersection. Do the same for the horizontal direction. Thus, we get 2 vanishing points.1. How do we use the projected points to find the 2D vanishing points?

I think not because if I compute the dot product of the above two vanishing points, I get a non-zero number.2. If we have two rays from the focal point to one vanishing point each, will they be orthogonal?

I have no idea on how to begin. How can we possibly find the focal length w/o having the previous 3D coordinates?3. Get the focal length.

I have a feeling this relates to computing the normal by using the cross product. Do I simply take two of the above lines (intersecting ones) in 2D, add a z-coordinate of 1 and then compute the normal?4. Get the normal to the plane of the rectangle.

Note that I did not put specific numbers or examples, because I'd like to learn it by doing the homework. If someone could please guide me into the correct direction, I'd appreciate it. Thanks!