Question

Let $z_1$ be a fixed point on the circle of radius $1$ centred at the origin in the argand plane and $z_1\ne \pm 1$. Consider an equilateral triangle inscribed in the circle with $z_1,z_2,z_3$ as the vertices taken in the counter clockwise direction. Then, $z_1{z}_2{z}_3$ is equal to

• A. $z_1^2$
• B. $z_1^3$
• C. $z_1^4$
• D. $z_1$

Answer 1

Answer: (B)

$z_1^3$

Given, $z_1\ne \pm 1$
Since, $z_2$ and $z_3$ can be obtained by rotating vector representing through $\theta =\frac{2\pi }{3}$ and $2\theta =\frac{4\pi }{3}$ respectively.
and, $\omega =e^{i\theta }and{\omega }^2=e^{i2\theta }$
$\therefore z_2=z_1\omega$ and $z_3=z_1{\omega }^2$
$\therefore z_1{z}_2{z}_3=z_1\times z_1\omega \times z_1{\omega }^2$
$=z_1^3{\omega }^3$
$=z_1^3$ $[\because {\omega }^3=1]$