Proving Orthogonality of Planes and Lines in 3D Geometry

[Curious3141]

They're all just lines.

Oh, 'X' means cross product. I thought you meant something else by FXG, etc., the lack of spacing threw me.

Then what you wrote is axiomatic, and it follows from the fact that we're working in 3 dimensions. Simple enough ?

 

[gnome]

I guess. It just bothers me that I can't put my finger on any specific definition or postulate that says that.

 

[SGT]

Draw a perpendicular from point w to the plane C. Any plane that contains that line is, by definition, perpendicular to C.
In particular, the plane that contains that perpendicular and the intersection of A and C. Since a point and a line define a plane, this plane must be plane A.
Now, consider the plane that contains the perpendicular and the intersection of B and C. By the same reasoning, this must be plane B.
So, the perpendicular from w to C belongs simultaneously to A and B and must be their intersection.
The intersection of A and B is orthogonal to all lines in C. QED.