Question

State whether the following statement is true or false.

The cube roots of unity 1, $\omega ,{\omega }^2$ are the vertices of an equilateral triangle

• A. True
• B. False

Answer 1

The correct option is A True

The cube roots of unity are $1,\omega ,{\omega }^2$. 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, $z_1=1,z_2=\omega ,z_3={\omega }^2$

Then,

$z_1^2+z_2^2+z_3^2=1+{\omega }^2+{\omega }^4=0$ and

$z_1{z}_2+z_2{z}_3+z_3{z}_1=1.\omega +\omega .{\omega }^2+{\omega }^2.1=\omega +{\omega }^3+{\omega }^2=1+\omega +{\omega }^2=0$

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.