First slide

Algebra of complex numbers

Question

Let $z$ and $w$ be two non-zero complex numbers such that $| z | = | w |$ and $\arg (z) + \arg (w) = π$ . Then $z$ equal

Moderate

A

$w$
B

$- w$
C

$w$
D

$- w$

Solution

Let $| z | = | w | = r$ and $\arg (w) = θ$ , so that
$\arg (z) = π - θ$
We have
$z = r [\cos (π - θ) + i \sin (π - θ)]$
$\begin{array}{l} = r (- \cos θ + i \sin θ) \\ = - r (\cos θ + i \sin θ) = - \bar{w} \end{array}$

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