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.
How often do you think most people need to know the advanced mathmatical properties of 3d space?
I think it is a shame that I'm a math student in university and needed to verify about such a thing. And if we're talking about people doing physics it might be even worst if they suck like me at 3d geometry.
Math students in university need to verify basically everything, that's a lot of what the career is about. I remember being humbled when asked to prove something as familiar to everybody as -1 * -1 = 1
hhhh abstract algebra and proof writing courses.