If $\cos \alpha +\cos \beta +\cos \gamma =0$ and $\sin \alpha +\sin \beta +\sin \gamma =0$, then find the value of $\cos (\alpha -\beta )$.

Solution

1. Let $z_1=e^{i\alpha }$, $z_2=e^{i\beta }$, and $z_3=e^{i\gamma }$.
2. The given equations imply $z_1+z_2+z_3=0$.
3. This means $z_1,z_2,z_3$ are vertices of an equilateral triangle inscribed in the unit circle.
4. The angle between any two vectors is $120$ degrees, so $|\alpha -\beta |=\frac{2\pi }{3}$.
5. Thus $$\begin{array}{rl}\cos (\alpha -\beta )& =\cos (120^{\circ })\\ & =-\frac{1}{2}\end{array}$$.
6. Hence the answer is (A).