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Properties of Modulus

a For any two...

Question

(a) For any two non-zero complex numbers $z_{1}$ and $z_{2}$ if $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ , then prove that $a r g z_{1} - a r g z_{2}$ is zero.
(b) Prove the above result if we have $| z_{1} - z_{2} | = | z_{1} | - | z_{2} |$

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Solution

(a) $| z_{1} + z_{2} | = \sqrt{r_{1}^{2} + r_{2}^{2} + {2 r}_{1} r_{2} cos (θ_{1} - θ_{2})}$
and $| z_{1} - z_{2} | = \sqrt{r_{1}^{2} + r_{2}^{2} - {2 r}_{1} r_{2} cos (θ_{1} - θ_{2})}$
If $cos (θ_{1} - θ_{2}) = 1$ i.e. $θ_{1} - θ_{2} = 0$
or arg $z_{1 -}$ arg $z_{2} = 0$ , then
$| z_{1} + z_{2} | = \sqrt{r_{1}^{2} + r_{2}^{2} + {2 r}_{1} r_{2}} = r_{1} + r_{2}$
$| z_{1} | + | z_{2} |$
and $| z_{1} - z_{2} | = r_{1} - r_{2} = | z_{1} | - | z_{2} |$

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