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You can visualize what I means for vectors to be perpendicular, you can draw this picture, alright? So you might say to yourself, I should be able to visualize this. I should be able to sit in a quiet room, turn lights off and visualise what it means for complex exponentials to be orthogonal. Yes, yes. I see it. No, you don’t. Let me relieve you with this burden. There is no reason in hell you should be able to visualize when two functions not let alone complex exponentials are orthogonal. Don’t beat yourself up trying to do that and don’t say to yourself that you are less of a person if you cannot visualise when two functions are perpendicular. Not only the complex I mentioned sines or cos-sines, you know, all sorts of, all sorts of innocent-looking functions are out there that you work with for all your life, turn out to have sort of orthogonality relationship, but you might say to yourself like sines or cos-sines, sin(nt) is the orthogonal of cos-sin (mt), sin 2π nt is the orthogonal of cos-sin (2π mt), also interesting results are like that be sure you might say to yourself Gee, if I look at the graph, I should be able to visualize this. Don’t bother, alright. There is no point, there is no point. It’s reasoning by analogy, alright. The fact is you establish these formulas, you establish the orthogonal and then you apply your intuition for orthogonal functions, or for orthogonal vectors you can visualise to situations where you can’t visualise it. All right, and that’s the real power of this line of reasoning because you’re gonna apply your intuition to places where it should have no business applying somehow, alright.