. 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