So I'm trying to prove a vector from the viewpoint of a pinhole camera to the vanishing point of a set of 3D parallel lines is parallel to the direction of the parallel lines. I figured proof by contradiction would help me understand all the geometric concepts together, but I feel like I'm missing a few steps:
Lets assume that a parallel set of 3D lines A,B, and C aren’t parallel to the vector from the viewpoint of a pinhole camera to the vanishing point. If so then there exists a point in which the vectors of the parallel lines touch the vector created from the view point of the pinhole camera to the vanishing point. If this is true, then there must be a point where the coordinates of the image plane and of the 3D world are exactly the same. But as this can't happen as the transformation between the two depends on the focal length and Z on the 3D plane, then the lines are parallel.