Question

The cube roots of unity when represented on the Argand plane form the vertices of an

• A. Equilateral triangle
• B. Isosceles triangle
• C. Right angled triangle
• D. None of these

Answer 1

The correct option is A Equilateral triangle

The cube roots of unity are $1,w,w^2$. We know that if $z_1,z_2,z_3$ are the vertices of an equilateral triangle in the Argand plane.Then $z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
If we take $z_1=1,z_2=w,z_3=w^2$
and $z_1{z}_2+z_2{z}_3+z_3{z}_1=1.w+w.w^2+w^2.1$
$=w+w^3+w^2=1+w+w^2=0$
Thus $z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
Therefore triangle is equilateral.
Note:Students should remember this question as a fact.