Geometrical Representation of Argument and Modulus
If |z1 + z2...
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(θ1−θ2)=1 ⟹θ1−θ2=0 ⟹arg(z1)=arg(z2) Ans: 1