Question

The cube roots of unity when represented on an Argand diagram form the vertices of an equilateral triangle. If this is true enter 1, else enter 0.

Answer 1

The cube roots of unity are
$1,w,w^2$
Where $w^3=1$
Now
If the form an equilateral triangle they must satisfy the following relation
$z_1^2+z_2^2+z_3^2-z_1.z_2-z_2.z_3-z_3.z_1=0$
Thus,
$1+w^2+w^4-w-w^2-w^3$
$=1+w^2+w^3.w-w-w^2-1$
$=(1-1)+w^2-w^2+(w-w)$
$=0$
Hence, they form an equilateral triangle.