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.

  • zaknenou@lemmy.dbzer0.comOPEnglish
    1·
    13 hours ago

    I tried again, I don’t find mistakes in your statements, I just don’t see how they make up for “instant in-mind proofs” for the problem I think I see it now, nevermind. Your got a very good visualization for 3D CanadPlus. It seems so intuitive that “the set of points that map to H with orthogonal projection is a straight line”, but do you happen to have a pocket proof for that ?