Plane intesection with vertical vector given plane normal

Question

There is a plane defined by a normal and an origin. For simplicity's sake, the origin is $(0,0,0)$.

And then, there are two coordinates ($x$ and $z$) of a point on this plane.

How can I find the third coordinate ($y$) of the point ?

Notes:

• The plane is never vertical, so the point always exists.
• This is exactly like finding the interection between the plane and an infinite vectical vector passing trough $(x,0,z)$
• In this space, the vertical (up) vector is $(0,1,0)$
• The computation needs to be cheap/fast.

Answer 1

Let $ax+by+cz=0$ be the equation of your plane with $\vec n=(a,b,c)$ being the normal vector. Clearly, the plane passes through the origin. Say you have a point $P(x_0,y_0,z_0)$ then
$$y_0=-(ax_0+cz_0)/b.$$