This may be a bit long winded but I have not done linear algebra in a while and I want to make sure my math is correct.
I am running two different computer simulations of proteins and I need to match them by mapping the peptide bond of one simulation onto another. For those who do not know much chemistry it looks like this:
All that really matters though is that I have three vectors all within one plane in both simulations. One from the C atom to the O atom, one from the C atom to the N atom, and one from the N atom to the H atom(white ball). I also have two different position vectors for a point, one for each simulation, P1 and P2, with the origin of the vector at C.
I will call these vectors, $\vec{CO}$, $\vec{CN}$ and $\vec{NH}$. Since there are two simulations both with these vectors, the other set will be $\vec{CO'}$, $\vec{CN'}$, and $\vec{NH'}$.
What I am doing is finding the cross product $\vec{CO}\times{\vec{CN}}$, calling it $\vec{w}$, and then finding the cross product $\vec{w}\times{\vec{CO}}$, and then normalizing all these vectors and putting them in a matrix $F$, and doing the same thing for the other set and calling it $H$. Then I am finding a transformation matrix by finding dot product of $H$ and the inverse of $F$.
$$ A\cdot{F}=H $$ $$ A=H\cdot{F^{-1}} $$
Once I have the transformation matrix $\vec{A}$ I am applying into to $\vec{P1}$ like this:
$$ A\cdot{\vec{P1}}=\vec{P1'} $$
and then measuring the distance between $\vec{P1'}$ and $\vec{P2}$. The problem I am having is that $\vec{P1'}$ is varying way too much between every step of the simulation. This is leading me to believe that I have done something wrong with the transformation matrix. Any help is appreciated and thank you in advance!