Question

Say true or false:

The cube roots of unity when represented on the Argand diagram form the vertices of an equilateral triangle.

• A. True
• B. False

Answer 1

The correct option is A True

The cube roots of unity are $1,w,w^2$
Now if they form an equilateral triangle, then it must satisfy the condition
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
$1^2+w^2+(w^2{)}^2$
$=1+w^2+w^3.w$
$=1+w^2+w$
$=0$
Now
$1.w+w.w^2+1.w^2$
$=w+1+w^2$
$=0$
Hence
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
Thus, the cube roots of unity form an equilateral triangle.