Mathematics Asked by Broly-29 on December 29, 2020

Given two complex numbers $z,w$ with unit modulus (i.e., $ |z|=|w|=1$), which of the following statements will always be correct?

a.) $|z+w|ltsqrt2$ and $|z-w|ltsqrt2$

b.) $|z+w|lesqrt2$ and $|z-w|gesqrt2$

c.) $|z+w|gesqrt2$ or $|z-w|gesqrt2$

d.) $|z+w|ltsqrt2$ or $|z-w|ltsqrt2$

It is a multiple choice question and only one option is correct.

My approach: As modulus of $z$ and $w$ is 1. Let $z=e^{ialpha_1}$ and $w=e^{ialpha_2}$. Now,

$$|z+w|= |e^{ialpha_1}+e^{ialpha_2}|$$

$$|z+w|=|2 cos(frac{alpha_1-alpha_2}{2})e^{frac{i(alpha_1+alpha_2)}{2}}|$$

$$|z+w|=2 |cos(frac{alpha_1-alpha_2}{2})|$$

I don’t know how to proceed after this. Any help would be appreciated.

(C) holds as a consequence of the parallelogram law:

$$ |z+w|^2 + |z-w|^2 = 2(|z|^2+|w|^2) = 4 $$ so that at least one of $|z+w|^2$ or $|z-w|^2$ must be $ge 2$.

I leave it to you to find counterexamples for (A), (B), and (D). These can easily be found by choosing $z, w$ from $-1, 1, i$.

Correct answer by Martin R on December 29, 2020

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