The triange $OAB$ in an Argand diagram is equilateral. $O$ is the origin and $A$ corresponds to the complex number $\sqrt{3}(1-i)$. B is represented by the complex number $b$.
Find the two possibilities for $b$ in the form $re^{i\theta}$. Illustrate the two possibilities for $OAB$ in a sketch.
Where I got up to:
I managed to convert $\sqrt{3}(1-i)$ into Euler's form to get $\sqrt{6}e^{\frac{-\pi}{4}}$ but after that I am completely lost on how to get $B$.
The point $B$ is obtained from $A$ rotating it by an angle of $\frac\pi3$ radians around the origin, clockwise or counter-clockwise. The first possibility leads to$$b=\sqrt3(1-i)\times\left(\frac12+\frac{\sqrt3}2i\right)=\frac{3}{2}+\frac{\sqrt{3}}{2}+\left(\frac{3}{2}-\frac{\sqrt{3}}{2}\right)i.\tag1$$For the other case, you use $\frac12-\frac{\sqrt3}2i$ instead of $\frac12+\frac{\sqrt3}2i$.
Besides, since $\sqrt3(1-i)=\sqrt6e^{-\pi i/4}$ and $\frac12+\frac{\sqrt3}2i=e^{\pi i/3}$, $(1)=\sqrt6e^{\pi i/12}$.