I'm wondering if we can speak of angles, specifically right angles when we live in an inner product space, but our product $\langle u, v\rangle$ is not the standard dot product that we all know and love, but rather some other obscure unknown product.
Specifically I'm looking at the first proof for Cauchy-Schwarz inequality proof found here https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
Basically we have 3 vectors $u,v, z \in \mathbb F^n$ such that $u = z + v$ and $\langle z, v \rangle = 0$
The author then goes on to say that this means $u, v, z$ form a right triangle and so we can use the pythagorean theorem.
It's not perfectly clear to me why this is true. Our inner product is obscure. It is not necessarily true that if $\langle z, v \rangle = 0$ then $ u \cdot v = 0$ right? And even if it did, in our world, the standard dot product "does not exist", so can we still use it to infer angles between vectors?
$u = z + v$, $\langle z, v \rangle = 0$. The standard dot product does not exist. Can we use the pythagorean theorem? why? What does right angle even mean in our sense?