How first reading felt:
How the second reading felt at the beginning:
How it ended up:
What is {y∈V | O(y) = 0} ? If the plane doesn’t pass through $0_V$ then how would that 0 be the image of some point ? Most likely you’re using something from linear algebra that I didn’t learn in my course (I didn’t learn projection I think, only examples when learning matrices).
DUH! If this was math.stackexchange I’d choose this as answer
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 ?
I couldn’t make sense of the first paragraph, are you sure it is right ?
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fyi: the orthogonal projection of a point P into a plane is a point H of that plane such that for any other point A of the plane: (PH) is orthogonal to (HA). One might think that finding that “(PH) is orthogonal to (HA)” for one such point A of the plane is enough, turns out it is not.
luckily an easier criterion exists: H is the orthogonal projection of P if (PH) is parallel to n the normal to the plane.
retro computing was so chad
ADHD driven hard work could never disappoint huh? 
But what was the advantage of QuickBasic? Weren’t C++ and Javascript around at the time? I only hear about them in this context
when I say forums, that includes math.stackexchange, please don’t call it shitpost, people there are really something to say the least.
hhhh abstract algebra and proof writing courses.
impressive, I’d like to ask abou stuff like how long it took you and stuff. But in this discussion I’d like to mention that I didn’t use any complicated terms, only orthogonal projection (middle school) and perpendicularity (elementary school).
my lazyass had it hard to put correct labels. But judging by how many people ignored the proble an are just scolding me for using AI, fair is fair.
transitive you mean ?
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.
3 years ago, a university teacher proposed it to me on facebook and added it to “the list”, but still didn’t go back to 
@TauZero@mander.xyz It is a Geogebra drawing I did to reason with the problem, I took a screenshot of the drawing to attached it.
In the drawing, the labels are different from the problem, but I just made a sphere whose diameter is [AP] (here point P has label A, while A has label A’),
then constructed the plane using A and two other points of the sphere (C and D in the picture),
I thought like “if that property from 2D geometry holds in 3D then any point in the intersection of plane and the sphere will satisfy the perpendicularity, and thus two of them will do for a counterexample”.
And It is exactly what happened: Using Geogebra’s tool of measuring angles it shows that the two points, C and D, that I picked up both satisfy the orthogonality condition (in the picture angle(A,C,A’)=90°=angle(A,D,A’), but they can’t be both the projection of P, right ? Counterexample! (the hypothesis was that a point on a plane that satisfy that condition is immediately THE projection of the point that isn’t on the plane)
Yeah It is not the best thing but I wanted to attach something, and the drawing that I used was the best thing at hand.
ChatGPT is trained based on forum discussions and pretty likely pirated books. If it found the idea in a previously established text it would have answered correctly. That’s why I DO think it is representative of what the average good student was taught (not how smart, or good at problem solving they be). What’s funny is that after reasoning it found the right answer, which is counter intuitive, since ChatGPT is supposed to be good at retrieving information, not at reasoning!
yeah, I’m starting all over again with university, so hopefully this will be eventually fixed. About the rest of the population though …
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.
I’m actually old and lurked in university stuff for a long time and dropped out of engineering in university and started with math all anew, yet at the same time I’m still a beginner.
I don’t exactly remember How I started thinking about the “distance between plane and a point formula”, I think I stumbled upon it while organizing my old bookmarks. Tried to make a proof, and in the process that question came, and when I couldn’t solve it on the fly I though like “it’s so over for me”. Then ChatGPT also got it wrong and was like “It’s so over for mankind”. And I ended up making this post to share my despair. Actually many answers were eye opening.