This question involves 3 dimensions, but I will start in 2 dimension to get a better fit explaination. Its been bothering me for a couple days and I've been unsuccesful as getting an simple answer (I am able to derive an answer, but it seems to complicated to be reasonable). So, 2D first:

Suppose I have a line and a parabola . I construct a vector from the origin to the point on . I want to determine the vector normal to at pointing towards . I know from the derivitive of that this vector, in general, is . Now, I need to determine a vector in the same direction as that when originating at the point will terminate exactly on the line (i.e. the x-axis). For this I understand I simply need to find a general value such that:

Lets say . And remember that . I know that, since I need the vector (originating at point I remind you) to terminate on the x-axis, must be of the form where is some general expression in terms of x So I need to find the value of such that:

Looking at P and V, I hope you can see that the value of would be , or I've done a poor job of explaining. This is because then we have:

This gives me the form of that assures it terminates on . But how would I determine a vector that terminates on when is not one of the axes? Say , how do I determine , originating at point such that it terminates on a line of this slope? Or a line of any slope?

And this question is the underlying question to the main one:

How would I determine this whole process for a parabaloid and a plane ? I know how to find the vector for but other wise I am lost, short of exiting vectors and determining line intersections, which will just defeat the purpose of my project. Anybody who can help, please do. I've been stumped on this one for some time. Thank you in advance