# Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors.

What I would like to know is angles (or real parts of the quaternions) by which I should rotate around given axis such that I would make the same rotation as original quaternion represents.

$$q=q1∗q2∗q3$$

Note: When the first rotation is being made, the second and third axis is being rotated also, and the second rotation affects the third axis in the same manner.

The solution I am looking for is the one which makes the least rotations, thus minimising rotations angles about the first, second and third axis.

• I think that there is some confusion in your question. Why you want the real parts? A quaternion representing a rotation is a pure imaginary quaternion. And, why you say that the first rotation acts on the other axis? If $p$ represent a rotation on avector $v$ than the rotated vector is $u=pvp^{-1}$ and if $p=p_1p_2p_3$ we have $u=p_1p_2p_3vp_3^{-1}p_2^{-1}p_1^{-1}$, so the first rotation performed is $p_3$ and this does not acts on the other rotations. – Emilio Novati Jun 2 '15 at 19:46
• Hey, thanks for your reply :). If we define a quaternion to be $q=w+xi+yj+zk$ where i,j,k are imaginary numbers of quaternion, then by real part I meant $w$. The reason for this is that I have an object which attitude can be manipulated by three motors and I know those motors axis of rotation. What I need is to figure out angles those motors should turn in order to achieve the same rotation as the original quaternion $q$. – Rytis Karpuška Jun 3 '15 at 7:45
• I think that you can start from some preliminary question, as here: math.stackexchange.com/questions/435680/…, or here math.stackexchange.com/questions/1175209/…. If you want a more complete treatment of the topics you can see: web.mit.edu/2.998/www/QuaternionReport1.pdf – Emilio Novati Jun 3 '15 at 9:24
• @EmilioNovati Rotation in $\Bbb R{\bf i}+\Bbb R{\bf j}+{\Bbb R}{\bf k}$ around a unit pure imaginary quaternion $\bf u$ by angle $2\theta$ is conjugation by ${\bf q}=\cos\theta+{\bf u}\sin\theta$, which certainly has a real part. – whacka Sep 14 '15 at 0:54