How do the algebraic and geometric formulas of dot and cross product relate to each other? I have seen the proof for 2 dimensions but how do we generalize it to $n$ dimensions? Is there a way to prove it without the cosine law? That proof feels like a chicken and egg proof as some use vectors to prove cosine rule and vice versa.
Proof for 2 dimensions: Proof of equivalence of algebraic and geometric dot product?