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.

  • bob_lemon@feddit.org
    3·
    17 hours ago

    AH and PH do form a right angle, that’s postulated in the problem. But P is only the projection of H onto the plane if PH is indeed parallel to n. Which is not necessary.

    Imagine a nail patrols hammered into a piece of wood at an angle. The wood surface is the plane, the entry point is H and the head of the nail is P. A is anywhere on the line perpendicular to the nail on the board.

    If you shine a light from above, you can see P’, the projection of P as the end of the shadow cast by thaw nail. Unless the nail is straight, P’ != H.