Question

If $|z_1|=|z_2|=|z_3|=1$ and $z_1,z_2,z_3$ are represented by the vertices of an equilateral triangle then

• A. $z_1+z_2+z_3=0$
• B. $z_1{z}_2{z}_3=1$
• C. $z_1{z}_2+z_2{z}_3+z_3{z}_1=1$
• D. None of these

Answer 1

Answer: (A)

$z_1+z_2+z_3=0$

Clearly $z_1,z_2,z_3$ lie on unit circle $|z|=1$

It is known that angle subtended at circumference must be half of that subtended at centre.

Hence the $3$ points are separated by ${120}^{\circ }$

So, $z_1+z_2+z_3=e^{i\theta }+e^{i(\theta +\frac{2\pi }{3})}+e^{i(\theta +\frac{4\pi }{3})}$

We know $e^{i\theta }=\cos \theta +i\sin \theta$

Using this expression $z_1+z_2+z_3$ gives value $0$.