Right angles in econometrics

I mentioned a week ago that the Generalised Method of Moments puts orthogonality relations (expressions stating that two vectors are at right angles to each other) at the centre of econometric estimation. In the spirit of advancement, I have tried to see whether perceiving econometric proofs and analysis in visual terms could be used more widely. It's not much of a study, only having lasted a week and concentrating on orthogonality.

Nevertheless, some of the results are encouraging. Considering the geometry behind the simple orthogonality relation a.b=0 (understandable by rotating axes and then using cos90=0) makes the two-way interaction between operations such as differentiation and the resulting econometric equations much clearer. I find it helps if in viewing the equations, one sees at the same time the proof of (a.b=0 is equivalent to perpendicularity). It's a trivial thing but helps a lot. The idea of differentiation as a projection also follows, and it is quite fun to imagine how the projected space gets rotated around at each successive differentiation.

The orthogonality approach also helps with the infinitesimal analysis which crops up in econometric proofs. The relation |x+y|<|x|+|y|, often used to show the term on the left hand side tends to zero by showing the two right hand ones do, is easy to show in Euclidean space (that's the one that describes our world) by projecting lines at right angles and using cos<=1. In more complex mathematical space, the equation seems to be equivalent (at least conceptually) to having projections shorter than the original line. So using the proof is like constructing a triangle and watching two of the sides contract, forcing the third one to contract.

I don't know if this approach will turn out to be an inefficient conceptual tool, but it avoids making proofs seem either repetitively drawn from someone else's work, or drawn magically from a bag of symbols. It is possible that leading mathematicians have the spatial perception buried in their subconscious, or they may work completely differently.