Generalised Scalar and Vector Products in \(\mathbb{A}^3\)

Anyone who wanders in here knows that if you have two vectors \(u\) and \(v\) in \(\mathbb{V}^3\) (the associated vector space of \(\mathbb{A}^3\)) the Euclidean scalar product \(u \cdot v\) and the Euclidean vector product \(u \times v\) are defined as usual and satisfies some properties. A big problem with some of these properties is that taking "lengths" of vectors is highly problematic; square roots are very hard to compute, let alone non-existent in finite fields. Moreover, "sines" and "cosines" of "angles" (whatever those terms mean) are not well-defined and require the use of techniques from integral calculus & (infinite) power series (specifically those of the Maclaurin flavour).

Let us now strip everything back and generalise our definitions of scalar and vector products so that we can apply what we have for a general field (not of characteristic \(2\)) and for arbitrary geometries.

Starting with \(\mathbb{V}^3\), the associated vector space of \(\mathbb{A}^3\), we equip a symmetric bilinear form on it and represent it by the matrix \( B \), so we can define such a symmetric bilinear form by

\( u \cdot_B v \equiv uBv^T. \)

This gives us a generalised scalar product, which we will call the \(B\)-scalar product. The Euclidean scalar product corresponds to the case when \(B\) is the \(3 \times 3\) identity matrix.

We can also define a generalised vector product on \(\mathbb{V}^3\) by

\( u \times_B v \equiv (u \times v) \text{adj}(B). \)

Here, we use \(u \times v\) to denote the Euclidean vector product. The generalised operation is called a \(B\)-vector product, and from here we can modify our usual results of vector products with respect to the matrix representation \(B\).

To end this note, we see that by stripping back all geometrical meaning our definitions of \(B\)-scalar products and \(B\)-vector products easily generalise to arbitrary geometries, parameterised by the symmetric bilinear form with matrix representation \(B\) (which gives different definitions of perpendicularity), as well as arbitrary fields not of characteristic \(2\).

Want more? Keep a look out for my PhD thesis, to be submitted in 4 weeks time!

Note by Gennady Notowidigdo
1 month ago

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Looking forward to it! Thank you for sharing Gennady!

Also, have you read "A new look at multi-set theory"(Wildberger) I was wondering if you felt confident enough to apply finite fields to this framework?(The one where he "creatively hijacks" mini-max algebra... to do away with the zero denominator problem).

I must say that the concepts of multi-set definitely needs to be made better use of to allow us to better communicate to each other.

Peter Michael - 3 weeks ago

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Thanks Peter! Yes, I have read Wildberger's paper on multisets; he has extended the framework over the past 1-2 years to make it more concise, but it's a very good start nonetheless. It's hard these days bouncing back and forth various research topics just to keep up appearances, so it might be a big ask for him to revisit this topic.

Gennady Notowidigdo - 3 weeks ago

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I would love to know how he extended this!!!

I can only imagine the fire juggling act he continues to do but if I could understand how to apply that framework to say chromogeometry, UHG, and rat-trig. I really think I could make a break-through on connecting the rubiks cube to modelling these theories to help young people(and myself, if I am being honest) to better understand the continuum and how to move towards helping others feel more confident in their understanding... on their own terms(without my preaching at them.)

My biggest goals is to have a course (mathematics) that allows people to explore just enough mathematics as they so desire with the help of a manipulative that they can explore, discover and understand for themselves. Without another person "yelling" at them.

The amount of progress that can be made would be... remarkable to say the least.

Any chance you can give me the "list" of this extension? Or where I could read about it?

Always great to hear from you!

Peter Michael - 3 weeks ago

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@Peter Michael Have a look at his YouTube channel; I think he introduced four types of data structures, each with clear specifications on what is and isn't allowed. I haven't looked into this that much; set theory interests me very little.

Always great to hear from you!

Given that I'm about to submit my PhD thesis in two weeks, thanks for the appreciation!

Gennady Notowidigdo - 3 weeks ago

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