This question is in the same spirit as this one.
On one hand it is often stated,that the Pythagorean theorem means that Euclidean distance arises from a scalar product. But on the other hand most elementary proofs rely on surface arguments.
Is there a way to make the connection explicit using linear algebra terms ? In other words, is there a version of the Pythagorean theorem of the form "if a norm satisfies this condition regarding determinants, then it is given by a scalar product ?"
Another way of looking at the question is this. $\mathbb{R}^2$ with euclidean norm, determinant, and scalar product, is a model for the Euclidean plane. This somehow means that linear algebra can give us some insight about geometry. Conversely, if the Euclidean plane is a model for a two-dimensional real space with determinants and norm, why must the norm be the one given by a scalar product ? What insight does Euclidean geometry give us about scalar products ?
One could argue that for the Pythagorean theorem angles are as important as surfaces, and that having angles implies a scalar product. But it seems to me the naïve conception of angles is very different from the abstract version of orthogonality in a scalar product space that we are used to.