Position of a plane that intersects point

Question

I'm trying to find the position of a plane such that it intersects an arbitrary point.

A point P exists along a vector M starting at O, a plane intersects P with a normal vector of N.

What is the position of P such that the plane intersects another point Q.

For the purposes of this problem you can think of O as being the origin and that both Q, M and N are known.

This problem has had me stumped for a while now and is vital to a personal project I'm working on. I'd like to understand the theory better so I don't get stuck like this again so any input as to what to study would be greatly appreciated.

Answer 1

The normal vector $N$ to the plane is known, and the point $Q$ which lies on the plane is known, therefore, the equation of the plane is

$ N \cdot (r - Q) = 0 $

Now point $P = \lambda M $ for some known vector $M$

Plug that in the equation of the plane,

$ N \cdot ( \lambda M - Q ) = 0 $

Solving for $\lambda$, one gets

$ \lambda = \dfrac{N \cdot Q} { N \cdot M } $

Now point $P$ is fully specified.