If there is no geometric meaning if the coordinates are not from an orthonormal basis, then where do we uses inner products for non-orthonormal basis? Or do we rather convert the coordinates into an orthonormal basis and then do the inner product to get a geometric meaning out of it?

goose_r_s

It sounds like most things play nice with an orthonormal basis. Is it a good idea then to wrap all operations into a process that first converts vectors to an orthonormal basis and then applies said operation?

tcarey

Is there a good visualization/intuition for why the dot product is meaningless without an orthonormal basis?

If there is no geometric meaning if the coordinates are not from an orthonormal basis, then where do we uses inner products for non-orthonormal basis? Or do we rather convert the coordinates into an orthonormal basis and then do the inner product to get a geometric meaning out of it?

It sounds like most things play nice with an orthonormal basis. Is it a good idea then to wrap all operations into a process that first converts vectors to an orthonormal basis and then applies said operation?

Is there a good visualization/intuition for why the dot product is meaningless without an orthonormal basis?