Why does the equivalence between the algebraic and geometric definitions of dot product relies on having a orthonormal basis? I do not get why they are only equivalent when we are working with a orthonormal basis, is the algebraic definition defined only for those basis? Thank you very much
 A: That’s basically it. In a non-orthonormal basis, the algebraic expression for the Euclidean scalar product—the geometric definition—also involves mixed terms, so it’s not a simple “dot product.” On the other hand, any two orthonormal bases are related via an orthogonal transformation matrix, and it’s pretty easy to show that the dot product is invariant under such transformations, so any orthonormal basis will do. Note, though, that there’s a bit of a circularity here because the notions of length and angle that are used in the geometric definition are tied to the choice of scalar product for the space.
A: I think you have to start with a Euclidean plane. Pick an origin and two unit vectors from the origin. The projection of each vector on the other has length the cosine of the angle between them by using right triangles. The dot product is a bi-linear functional on two vectors. The geometric definition forces the dot product of two perpendicular vectors to be zero and of a unit vector with itself to be one, and this gives the algebraic definition.
