I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)?(adsbygoogle = window.adsbygoogle || []).push({});

I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say [itex]\boldsymbol{\sigma_1}[/itex]] ONTO another direction-vector [say [itex]\hat{a}[/itex]] FOLLOWED BY an inner product of the now-aligned vectors. The randomness follows from this fact: I am here treating the "spun" vector (the one rotated) as an unknown (ie, hidden) direction-vector associated with complex dynamics.* By way of example:

Let: [itex]\boldsymbol{\sigma_1}+\boldsymbol{\sigma_2}=0.[/itex] (1)

Given: [itex]\hat{a}\circ\boldsymbol{\sigma_1}=+1[/itex]; (2)

Then: [itex]\hat{a}\circ\boldsymbol{\sigma_2}=-\hat{a}\circ\boldsymbol{\sigma_1}=-1[/itex]. (3)

In general: [itex]\hat{a}\circ\boldsymbol{\sigma}=\pm\hat{a}.\hat{a}

=\sigma_{\hat{a}}=\pm1[/itex]. (4)

Expectations: [itex]\left\langle \hat{a}\!\circ\!\boldsymbol{\sigma}\right\rangle =0;

\left\langle (\hat{a}\!\circ\!\boldsymbol{\sigma})^{2}\right\rangle =1

[/itex]. (5)

Question: Does such a vector-product exist in GA? More cheekily: Should it?

--------------------------

* The proposed "spin-product" allows all sorts of interesting combinations: eg, combine random or non-random rotation-and-reduction, rotation-only, rotation-and-dilation of a vector onto another followed by another vector-product of the now-aligned vectors; etc.

Clearly, a user of this product needs to specify the variant being utilised; maybe with variant spin-symbols like:

a[itex]\circ[/itex]b, a_{[itex]\circ[/itex]}b, a^{[itex]\circ[/itex]}b, a[itex]\bullet[/itex]b; etc. But I digress!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I A new vector-product for geometric algebra?

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**