I think 3D geometry has a lot of quirks and has so many results that un_intuitively don't hold up. In the link I share a discussion with ChatGPT where I asked the following:
assume a plane defined by a point A=(x_0,y_0,z_0), and normal vector n=(a,b,c) which doesn't matter here, suppose a point P=(x,y,z) also sitting on the space R^3. Question is:
If H is a point on the plane such that (AH) is perpendicular to (PH), does it follow immediately that H is the projection of P on the plane ?
I suspected the answer is no before asking, but GPT gives the wrong answer "yes", then corrects it afterwards.
So Don't we need more education about the 3D space in highschools really? It shouldn't be that hard to recall such simple properties on the fly, even for the best knowledge retrieving tool at the moment.
Wait is that not true? Why wouldn't H form a right angle with P and A?
AH would be perpendicular to n, and PH would be parallel to n, making them perpendicular to each other? Or am I misunderstanding the definition of a plane projection?
if (PH) is perpendicular to (AH) and n is perpendicular to (AH) ==> it doesn't really follow that (PH) is parallel to n, unlike in 2D geometry. ChatGPT also got the wrong implication at first.
Props to you for being one the few comments who actually understood the problem from my horrible statement/language though.