Find the rotation from two sets of 3 vectors?

Question

I have three linearly independent vectors ($\vec{a}$, $\vec{b}$, and $\vec{c}$) which have been rotated to three other linearly independent vectors $\vec{a}'$, $\vec{b}'$, and $\vec{c}'$. I would like to find this rotation.

I've looked at quaternions but each pair of vectors $\vec{a}$ and $\vec{a}'$ yields different quaternions, non of which are correct.

For example, let $\vec{a} = (1, 1, 4)$, $\vec{b} = (4, 1, -1)$, $\vec{c} = (-5, 17, -3)$, $\vec{a}' = (2.760834, 1, 3.062319)$, $\vec{b}' = (3.062319, 1, -2.760834)$, and $\vec{c}' = (-5.823153, -17, -0.301485)$.

Using this method, I get $q_{a} = 2.91634 + -0.160763i + 1.36833j + -0.301891k$, $q_{b} = 2.91634 + -0.301891i + 1.36833j + 0.160763k$, and $q_{c} = 12.6495 + 1.8133i + 0.630935j + 0.553128k$.

The real rotation is 28 degree in the y axis.

My question is, is there a way of finding the angle of rotation and the axis of rotation, preferably in the form of quaternions but I don't mind, of these three vectors?

Answer 1

You can obtain the rotation matrix (and from this, the axis rotation and the angle of rotation), by solving the sistem

$R\cdot(\vec{a},\vec{b},\vec{c})=(\vec{a}',\vec{b}',\vec{c}')$

That is:

$R=(\vec{a}',\vec{b}',\vec{c}')(\vec{a},\vec{b},\vec{c})^{-1}$

Here $R$ is the rotation matrix, and $(\vec{a},\vec{b},\vec{c})$ is the matrix with the three vectors as columns