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In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects"

So with that in mind, is there any geometric connection between two orthoganal functions on some interval? When I look at a plot of say, the legendre polynomials, nothing pops out at me as hinting there may be some orthoganality going on. Is there anything to see or do I have to look somewhere else ( like the integrals of the polynomials)? Thanks!