geometry rotation quaternion

Question

Express the rotation of $\mathbb R^3$ by $\frac{\pi}{4}$ about the $x = y,\ z = 0$ axis by using quaternions and identifying $\mathbb R^3$ with $(i, j, k)$-space. Find the image of the point $(\frac{1}{\sqrt 2}, \frac{-1}{\sqrt 2}, 0)$ under this rotation.

Answer 1

The rule is: take a unit imaginary quaternion $\omega$ and $v$ an imaginary quaternion. Then conjugation of $v$ by $q=e^{\omega t}=\cos(t) + \omega\sin (t)$, $v\mapsto qv\bar q$, rotates $v$ around the $\omega$ axis by angle $t/2$. So in your case, take $\omega=(i+j)/{\sqrt 2}$, $t=\pi/2$, $v=(i-j)/{\sqrt 2}$.