Question

If the following numbers $z_1,z_2$ and $z_3$ represent the vertices of an equilateral triangle such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$, then $z_1+z_2+z_3=0$. If this is true enter 1, else enter 0.

Answer 1

Let $z_1=w$ $z_2=w^2$ and $z_3=1$
Since we know that cube roots of unity form a equilateral triangle.
Now we know that
$w^3=1$ and $1+w+w^2=0$
Hence, $z_1+z_2+z_3=0$