I have 6 vectors in 3D space, namely p,q,a,b,c,d. For example p can be p=(0,0,1). I would like to solve the following equations to find R given these 6 vectors:

R(-p)R’ = p (where R’ is R_dagger)

R(-q)R’ = q

…

R(-d)R’ = d

which can be multiplied to give 1 equation, namely:

RpqabcdR’ = pqabcd

Further:

Rpqabcd = pqabcdR

[R,pqabcd] = 0 (where the commutator is defined [A,B] = AB - BA)

I am a bit confused since this equation is solved by R = 1 (1 being the identity by setting θ = 0), but the individual equations cannot be solved by R = 1 since R(-p)R’ = p gives -p = p ie all vectors would have to be 0.

Can I use R = pqabcd?

If so how can I compute this efficiently and fast computationally?

The point of this is that if an R can be found then the set of p,q,a,b,c,d represent a non-chiral set and if R does not exist then the set is chiral. Basically, we start from the set and apply the parity operator which gives a minus sign to all the vectors in the set and try to find a single rotation to bring them all back to the original set. It is like looking at your right hand in the mirror and checking to see if it can be super-imposed on your real right hand - it cant.

Thanks,

Takis