Given two complex numbers $z,w$ such that $|z|=|w|=1$. Find the correct statement.

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

(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

Related Questions

Proof of cyclicity using power of a point

1  Asked on November 22, 2020 by david-dong

Is a square a special case of a diamond (rhombus)?

2  Asked on November 21, 2020 by aspro

How to compute sum of sum of gcd of factor pairs of a number upto a large number efficiently?

2  Asked on November 21, 2020 by manas-dogra

Every $mathbb{P}-$integrable function $uinmathcal{L}^1(mathbb{P})$ is bounded (Counterexample)

1  Asked on November 21, 2020 by xenusi

Elementary Inequality regarding Sum of Squares

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

1  Asked on November 20, 2020 by janine

Implicit function theorem for $f(x,y,z)=z^2x+e^z+y$

1  Asked on November 19, 2020 by lena67

Complex Analysis Are My Answers Correct?

1  Asked on November 19, 2020 by nerdy-goat

integer programming with indicators

1  Asked on November 18, 2020

Is Hilbert’s tenth problem decidable for degree $2$?

2  Asked on November 17, 2020

It is sufficient to find the inverse of an element in a subgroup?

1  Asked on November 17, 2020 by kapur

Can we solve the following ODE via reduction of order? $y”+y”’sin(y)=0$.

1  Asked on November 16, 2020 by trevor-frederick

If a number can be expressed as sum of $2$ squares then every factor it can be expressed as sum of two squares

0  Asked on November 15, 2020 by soham-chatterjee

Does pairwise independence and same distribution imply trivial Invariant $sigma$-algebra?

1  Asked on November 14, 2020 by davi-barreira

An inequality involving positive real numbers

1  Asked on November 14, 2020 by jcaa

how to convince myself cos(A-B) proof works for all positions of A and B?

1  Asked on November 13, 2020 by across

Divergence of a function product with functions in $L^infty (Omega)$ and $H^1(Omega)$

2  Asked on November 12, 2020 by vitagor

Area of a triangle using determinants of side lengths (not coordinates of vertices)

0  Asked on November 12, 2020 by rohitt