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.

Mathematics Asked by Broly-29 on December 29, 2020

1 Answers(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

1 Asked on November 22, 2020 by david-dong

contest math euclidean geometry geometry power of the point solution verification

2 Asked on November 21, 2020 by aspro

2 Asked on November 21, 2020 by manas-dogra

algorithms arithmetic functions computational mathematics elementary number theory gcd and lcm

1 Asked on November 21, 2020 by xenusi

1 Asked on November 21, 2020 by consider-non-trivial-cases

1 Asked on November 19, 2020 by lena67

1 Asked on November 19, 2020 by nerdy-goat

1 Asked on November 18, 2020

2 Asked on November 17, 2020

decidability diophantine equations elementary number theory reference request

1 Asked on November 17, 2020 by kapur

1 Asked on November 16, 2020 by trevor-frederick

1 Asked on November 15, 2020 by lord-shadow

0 Asked on November 15, 2020 by soham-chatterjee

0 Asked on November 15, 2020 by ali-shadhar

closed form harmonic numbers integration polylogarithm sequences and series

1 Asked on November 14, 2020 by davi-barreira

1 Asked on November 14, 2020 by jcaa

1 Asked on November 13, 2020 by across

2 Asked on November 12, 2020 by vitagor

distribution theory partial differential equations sobolev spaces weak derivatives

0 Asked on November 12, 2020 by rohitt

Get help from others!

Recent Answers

- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed

Recent Questions

- Why fry rice before boiling?
- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?

© 2022 AnswerBun.com. All rights reserved.