Asklemmy

While 3D geometry is more difficult for me than 2D, I could almost immediately tell that the answer is no, there are infinitely many points H that satisfy this. The reason it's unintuitive is that our intuition about what "perpendicular" means comes from 2D and poorly translates to 3D.

The most intuitive explanation I can muster is this: imagine all possible planes that pass through both A and P. It should be obvious that there are infinitely many of them (I visualize it as a plane "rotating" around the AP axis). Each of these planes intersects the given plane since it passes through A. Think of the intersection line. It never passes through P (unless P is on the plane), so it is always possible to draw a perpendicular line from P to that intersection line. With one exception (when the perpendicular line falls on the A point), the point where the perpendicular falls satisfies the conditions for H. (I think all such points actually form a circle with AP' as the diameter, where P' is the parallel projection of P to the given plane, but I'm not 100% sure)

Well see, here you have good proof that chatGPT isn't actually "the best knowledge retrieving tool at the moment". ChatGPT (and every other LLM) suuuucks at complicated math, because these text extruders don't reason. Seriously, try out some more complicated math problems. I think you'll find chatGPT gets most of them wrong, and in infuriating ways that make very little sense.

I don't disagree that we need better math instruction for students. I've been saying this since I was a student. But using chatGPT being horrible at math as evidence of this is, well, ridiculous, frankly. ChatGPT's performance isn't based on how well your average high schooler understands something, and I don't know why you're trying to tie those two very different things together.

I believe this has multiple reasons in education. Most 3D stuff is assumed to be self-taught later by the interested learner in private somehow.

• the cross product for 3D is introduced within 1 day. No tasks or assignment where given to me that improve my intuition and proofs on its properties. a × b. Tbh the Cartesian product and determinants of 3x3 matrices are way more important than the time dedicated to them in average education systems.
• rotation in 2D space is very intuitive "clockwise" and "counterclockwise" everyone learns that at age 11 minimum. While in 3D you suddenly need gimbals and f**** quarternions, sphere coordinates, which are also never properly explained to the average person. I needed to teach myself why sphere coordinates work the way they do, my teachers think it was obvious I think. Coding something up in a 3D game engine helps with this immensely imo.
• right hand rule and its connection to the cross product. Its connection to physics and magnetic fields, forces that work on every charged particle in our universe. 🌌

Personally I am interested in a concept I call prime spaces. This to me means an intersection of geometry and number theory. Every entry in Matrix or Vector has to be a prime number. Geometry is connected to every other field of math.