First slide

Algebra of complex numbers

Question

If $z_{1} + z_{2} + z_{3} = 0$ and $|z_{1}| = |z_{2}| = |z_{3}| = 1,$ then value of $z_{1}^{2} + z_{2}^{2} + z_{3}^{2}$ equals

Moderate

A

-1
B

0
C

1
D

3

Solution

$\begin{array}{l} z_{1}^{2} + z_{2}^{2} + z_{3}^{2} \\ = {(z_{1} + z_{2} + z_{3})}^{2} - 2 (z_{2} z_{3} + z_{3} z_{1} + z_{1} z_{2}) \\ = 0 - 2 z_{1} z_{2} z_{3} (\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}) \\ = - 2 z_{1} z_{2} z_{3} ({\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3}) \\ = - 2 z_{1} z_{2} z_{3} (0) \\ = 0 \end{array} [∵ |z_{1}| = |z_{2}| = |z_{3}| = 1]$

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