Question

If the complex numbers $z_1,z_2,z_3$ represents vertices of an equilateral triangle such that $|z_1|=|z_2|=|z_3|$, then which of following is correct?

• A. $z_1+z_2+z_3\ne 0$
• B. $Re(z_1+z_2+z_3)=0$
• C. $lm(z_1+z_2+z_3)=0$
• D. $z_1+z_2+z_3=0$

Answer 1

Answer: (B), (C), (D)

$Re(z_1+z_2+z_3)=0$
$lm(z_1+z_2+z_3)=0$
$z_1+z_2+z_3=0$

$|z_1|=|z_2|=|z_3|$

where $z_1,z_2,z_3$ are the vertices of a equilateral triangle.

Let us assume

$z_1=r{e}^{i\pi },$ $z_2=r{e}^{i\frac{\pi }{3}},$ $z_3=r{e}^{i\frac{-\pi }{3}}$

So, $z_1+z_2+z_3=0$

Also, $Re(z_1+z_2+z_3)=(-1+\frac{1}{2}+\frac{1}{2})=0$

$Im(z_1+z_2+z_3)=(0+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2})=0$