Rotating a vector combinations

Question

Let's say I have two combinations of three vectors $A$, $B$, and $C$, and $A'$, $B'$, and $C'$ (and $AB = A'B'$, $BC = B'C'$, $BC = B'C'$). What is the way to rotate $A$, $B$, $C$ such that it superimposes on $A'$, $B'$, $C'$. I want to know the angle of rotation and axis of roation.

Answer 1

$A-A'$, $B-B'$ and $C-C'$ are all orthogonal to the axis of rotation, so you can form the cross product of two of them to find its direction. Then you can project everything along that direction into two dimensions, where you can use complex numbers. You're then looking for a rotation, represented by a complex number $z$, and an origin, represented by a complex number $x$, such that

$$ z(w-x)=w'-x\;, $$

where $w$ represents $A$, $B$ and $C$. You can eliminate $zx-x$ from these equations and solve one of the resulting equations for $z$, then substitute to solve for $x$.

This is of course all assuming that your calculations are exact. If you actually have data with errors, you might want to do a fit to find the best rotation. I'm sure there are lots of explanations of how to do that on the Web.