What does multiplying two quaternions result in?

Question

My quaternion is in the form just to be clear. (c,sx,sy,sz)

So my first quaternion is. Q=(0.966,0,0,0.966) and second is P=(0.966,0,0,0.966)

Which is a angle of 30 rotated around the z axis. The above I got from formula (cos(a/2),sin(a/2)N)

So my question is, what would be Q*P. multiplying two quaternions results in a rotation right?

So if I were to convert to complex(since I'm going around the z axis) wouldn't the answer be

0.5 + 0.866i

???

Answer 1

A quaternion is a number of the form $q=q_0 + q_1i+q_2j+q_3k$ where $i^2=j^2=k^2=ijk=-1$.

You can also write quaternions in polar notation like $q=|q|e^{\theta \hat n}$ where $\hat n$ is a unit pure imaginary quaternion. The quaternions of the form $q=e^{\theta \hat n}$ are important because they represent rotations about the axis $\hat n$ by an angle of $2\theta$ (the two comes from the way you actually use these quaternions to rotate vectors).

A quaternion which represents a rotation of $30^\circ$ around the $z$ axis is given by $$q=e^{\pi k/12} = \cos(\pi/12) + \sin(\pi/12)k \approx 0.966 + 0.26k$$

Multiplying two unit quaternions has the effect of rotating around one axis and then another. For instance if $q=e^{\theta_1\hat n_1}$ and $r=e^{\theta_2\hat n_2}$, then $qr$ would represent a composition of rotations: first rotate your vector by $\theta_1$ around the axis $\hat n_1$ and then rotate your vector by $\theta_2$ around the axis $\hat n_2$.

You should note however that rotations in $\Bbb R^3$ are not commutative in general. That is rotating via $q$ first and then $r$ will usually not result in the same vector as rotating via $r$ first and then $q$.