TITLE: Quaternions: Why is the angle $\frac{\theta}{2}$?
QUESTION [7 upvotes]: The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta 2\right)$$ $$w' = \cos\left(\frac \theta 2\right)$$ But why $\frac \theta 2$? Why not just $\theta$?

REPLY [5 votes]: Hint:
Consider the unitary quaternion 
$$
u=\cos \theta + \bf{i}\sin \theta 
$$
and the pure imaginary quaternion ${\bf j}$.
Then the product:
$
u{\bf j}u^*
$ (where $u^*=\cos \theta - {\bf i}\sin \theta$ is the conjugate of $u$) is:
$$
(\cos \theta + {\bf i}\sin \theta){\bf j}(\cos \theta - {\bf i}\sin \theta)=
(\cos \theta + {\bf i}\sin \theta)({\bf j}\cos \theta + {\bf k}\sin \theta)=
$$
$$
=(\cos^2 \theta -\sin^2 \theta){\bf j}+2{\bf k}\cos \theta \sin \theta = {\bf j}\cos 2 \theta+{\bf k} \sin 2 \theta
$$
that is the vector ${\bf j}$ rotated by an angle $2 \theta$ around the ${\bf i}$ axis.
Now, do the same for the other axis ${\bf j}$ and ${\bf k}$ and extend this result to any rotation around an axis ${\bf u}= u_1{\bf i}+u_2{\bf j}+u_3{\bf k}$.