Geometry with a general dot product | WildTrig: Intro to Rational Trigonometry | N J Wildberger

With the powerful point of view of rational trigonometry, metrical geometry can be extended far beyond even the blue, red and green chromogeometry of the last few lectures. We can consider a general dot product given by an arbitrary (invertible) symmetric 2x2 matrix, and consider corresponding notions of quadrance, perpendicularity and spread.

To illustrate this, we show how it works in a particular case, where the unit circle is an ellipse in our usual view. This powerful general approach will turn out to have major applications to classical triangle geometry.

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[Igdrazil]

You definitaly diserve a Fields Medal, at least for your pedagogical work "Make Mathematics Great Again", which is always brilliant, deep and accessible, rooted and iconoclast, generous and suggestive, and all at once inspiring for Freedom of Science and indépendance of thoughts. Strangely enought such distinction for such crucial spine bone of Knowledge, is still to be created. Here are certainly some inspireing elements for such futur fundamental realisation

[cbeezy]

I can't help but notice that using a generalize dot product in this way strongly resembles the way operators work on inner products in a hilbert space, for example, the way the "momentum operator" modifies the conjugate wavefunction before they are multiplied under the inner product. Is there a connection here?