Question

The cube roots of unity

• A. lie on the circle $\left|z\right|=1$
• B. are collinear
• C. form an equilateral triangle
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A lie on the circle $\left|z\right|=1$
C form an equilateral triangle

We know that cube roots of unity $1,\omega ,{\omega }^2$ satisfy $\left|z\right|=1$

Also, ${|1-\omega |}^2={\left(\frac{3}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2=3\Rightarrow |1-\omega |=\sqrt{3}$

$|\omega -{\omega }^2|=\left|\sqrt{3}i\right|=\sqrt{3}$

and ${|1-\omega |}^2=|1-(-\frac{1}{2}-\frac{i\sqrt{3}}{2})|=|\frac{3}{2}+\frac{i\sqrt{3}}{2}|=\sqrt{3}$

Therefore $1,\omega ,{\omega }^2$ form an equilateral triangle.