A question about triangle inequality in complex plane

Sunday, 6 March 2016

When does equality hold in the triangle inequality
$$|z_1+z_2+\dots+z_n|\leq|z_1|+|z_2|+\dots+|z_n|$$

Interpret your result geometrically

how to solve this problem i am really don't have please help me with this

for n=2
$|z_1+z_2|^2=|z_1^2+2Re(z_1\overline{z_2)}+|z_2|^2\\
\leq |z_1|^2+2z_1\overline{z_2}+|z_2|^2$

hence $|z_1+z_2|\leq|z_1|+|z_2|$

Answer

Hint:)

For $n=2$ use ${\bf Re\,}(z_1\overline{z_2})\leqslant|z_1||z_2|$ and by induction prove the general case!

For geometrically result, consider $z_i$ as a vector and see parallelogram rule to obtain geometrically consideration.

here $z=z_1+z_2+z_3+z_4+z_5$.

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