Laser Plane Estimation for Laser-Camera system?

I have to set up a system of a laser line/ plane projector and a web camera to localize the 3D position of the laser in the camera image. I’ve read / come across several resources but the idea is still not quite concrete in my head.

So my intuition is that since we have a set up of a laser projector and the camera, and we want to find the position of the laser point in the image – we have to find the ‘correct’ laser plane that intersects with the camera / image plane. I am confused as to how we find the relative pose of this plane with respect to the camera, and how we can use this to find the 3D coordinates?