A new vector-product for geometric algebra?

[N88]

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)?

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 $\boldsymbol{\sigma_1}$] ONTO another direction-vector [say $\hat{a}$] 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: $\boldsymbol{\sigma_1}+\boldsymbol{\sigma_2}=0.$ (1)
Given: $\hat{a}\circ\boldsymbol{\sigma_1}=+1$; (2)
Then: $\hat{a}\circ\boldsymbol{\sigma_2}=-\hat{a}\circ\boldsymbol{\sigma_1}=-1$. (3)

In general: $\hat{a}\circ\boldsymbol{\sigma}=\pm\hat{a}.\hat{a}<br /> =\sigma_{\hat{a}}=\pm1$. (4)
Expectations: $\left\langle \hat{a}\!\circ\!\boldsymbol{\sigma}\right\rangle =0;<br /> \left\langle (\hat{a}\!\circ\!\boldsymbol{\sigma})^{2}\right\rangle =1$. (5)

Question: Does such a vector-product exist in GA? More cheekily: Should it?
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* 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$\circ$b, a_$\circ$b, a^$\circ$b, a$\bullet$b; etc. But I digress!

 

[jedishrfu]

GA meaning Genetic Algorithms?

Oops got it Geometric Algebra.