What is the necessary and sufficient condition for $|z_1+z_2+\cdots+z_n|=|z_1|+|z_2|+\cdots+|z_n|,$ where $z_1,z_2,\cdots,z_n$ are any complex numbers.

  • $\begingroup$ I think all those complex numbers must be collinear! $\endgroup$ – user159888 Nov 25 '16 at 17:35

In triangle inequality $|z_1+z_2|\leq |z_1|+|z_2|$ we have equality iff $z_1=0$ or there is a $\lambda\geq 0$ with $z_2=\lambda z_1$.

So $|z_1+\dots+ z_n|=|z_1|+\dots+ |z_n|$ iff there are $i\in \{1,\dotsc,n\}$ and $\lambda_j\geq 0, j \in \{1,\dotsc,n\}\setminus \{i\}$ with $z_j=\lambda_jz_i$ for all $i\neq j.$

  • $\begingroup$ Thank you. All $z_i's$ must be on a line. $\endgroup$ – user159888 Nov 25 '16 at 17:39
  • $\begingroup$ And they must have "the same direction". $\endgroup$ – user302982 Nov 25 '16 at 17:41
  • $\begingroup$ Yes. Thank you. But is there any special significance if the line containing all these $z_i's$ pass through the origin!? $\endgroup$ – user159888 Nov 25 '16 at 17:46
  • 1
    $\begingroup$ It must be a line through the origin! $\endgroup$ – user302982 Nov 25 '16 at 17:53

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