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Question

If |z1+z2|=|z1|+|z2|, then show that arg(z1)=arg(z2).

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Solution

|z1+z2|= |z1|+|z2|
or |z1+z2|2=|z1|2+|z2|2+2|z1||z2|
or |z1|2+|z2|2+2Re(z1¯¯¯z2)=|z1|2+|z2|2+2|z1||z2|
or 2Re(z1¯¯¯z2)=2|z1||z2|
cos(θ1θ2)=1
θ1θ2=0
arg(z1)=arg(z2)
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