Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1 – z2) = 0.

Question

State True or False:

Let z₁ and z₂ be two complex numbers such that |z₁ + z₂| = |z₁| + |z₂|, then arg (z₁ – z₂) = 0.

Answer 1

False

Explanation:

Given |z₁ + z₂| = |z₁| + |z₂|

⇒ |z₁ + z₂|² = |z₁|² + |z₂|² + 2 |z₁| |z₂|

⇒ |z₁|² + |z₂|² + 2 Re (z₁z̅ ₂) = |z₁|² + |z₂|² + 2 |z₁| |z₂|

⇒ 2 Re (z₁ z̅ ₂) = 2 |z₁| |z₂|

⇒ cos (θ₁ – θ₂) = 1

⇒ θ₁ – θ₂ = 0

∴ arg (z₁) – arg (z₂) = 0