First slide

Algebra of complex numbers

Question

If $|z_{1}| = |z_{2}| = |z_{3}| = 1$ and $z_{1} + z_{2} + z_{3} = \sqrt{2} + i$ then the number $z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1} is:$

Moderate

A

a positive real number
B

a negative real number
C

always zero
D

a purely imaginary numbe

Solution

$| \sqrt{2} + i |^{2} = {|z_{1} + z_{2} + z_{3}|}^{2}$
$\Rightarrow 3 = {|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2} + 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1})$
$\Rightarrow 3 = 1 + 1 + 1 + 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1})$
$\Rightarrow Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) = 0$

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