> You end up needing trigonometric angle formulae that are at least as hard to prove as Pythagoras's theorem.
(emphasis mine)
Well that's not wrong (because proving Pythagoras' theorem is pretty straightforward anyway) but at the same time the one trigonometric formula you need (cos(a-b) = cos(a)cos(b)+sin(a)sin(b)) “follows from a (b+c) = ab + ac” if you start from Euler's formula.
It's so straightforward that you've used the exponential function on the complex plane to prove it?! You started by claiming that Pythagoras's theorem follows "directly" from the definition of the dot product. Can you admit that we're now quite far away from that?
To be honest, your comments have been quite low effort. They amount to "yeah but that bit's pretty easy too" while leaving it to me to work out how (and whether) your points fit into a coherent proof. I do get why this stuff all feels so trivial: we usually skip over it in higher-level proofs. But the only reason we can is that we can use nice abstractions like the dot product with its equivalent definitions, and that's thanks to the foundation these lower-level theorems provide.
> It's so straightforward that you've used the exponential function on the complex plane to prove it?! You started by claiming that Pythagoras's theorem follows "directly" from the definition of the dot product. Can you admit that we're now quite far away from that?
I does follow from the definition and common properties of the dot product, which was my original point. But you claimed it was circular because these properties derived from Pythagora's theorem, and so we've ended up showing it doesn't need to. And this later part was obviously much more involved than just “using the dot product”. But that's as if we had to prove that real numbers' multiplication is actually distributive over addition when saying “it follows from a (b+c) = ab + ac”, it's far from trivial if you want to go this far…
> To be honest, your comments have been quite low effort. They amount to "yeah but that bit's pretty easy too" while leaving it to me to work out how (and whether) your points fit into a coherent proof. I do get why this stuff all feels so trivial: we usually skip over it in higher-level proofs. But the only reason we can is that we can use nice abstractions like the dot product with its equivalent definitions, and that's thanks to the foundation these lower-level theorems provide.
I agree with you here, even on the low-effort part, I'm not particularly comfortable writing math on a keyboard and especially not on an English speaking forum because the notations are very different than the ones we use in France.
(emphasis mine)
Well that's not wrong (because proving Pythagoras' theorem is pretty straightforward anyway) but at the same time the one trigonometric formula you need (cos(a-b) = cos(a)cos(b)+sin(a)sin(b)) “follows from a (b+c) = ab + ac” if you start from Euler's formula.