<p>If <span>\(z_1\)</span> and <span>\(z_2\)</span> are two non-zero complex numbers such that <span>\(|z_1+z_2|=|z_1|+|z_2|\)</span> then <span>\(\arg z_1 - \arg z_2\)</span> is equal to</p>
Step-by-Step Solution
Key Concept: The equality |z₁+z₂| = |z₁|+|z₂| holds if and only if z₁ and z₂ lie on the same ray from the origin, meaning they have the same argument (same direction in the complex plane).
<p><strong>Step 1:</strong> Recall the triangle inequality for complex numbers: |z₁+z₂| ≤ |z₁|+|z₂|, with equality if and only if z₁ and z₂ have the same argument.</p><p><strong>Step 2:</strong> Given that |z₁+z₂| = |z₁|+|z₂| (equality holds), we must have arg(z₁) = arg(z₂).</p><p><strong>Step 3:</strong> This means z₁ = r₁e^(iθ) and z₂ = r₂e^(iθ) where r₁, r₂ > 0 and θ = arg(z₁) = arg(z₂).</p><p><strong>Step 4:</strong> Therefore, arg(z₁) - arg(z₂) = θ - θ = 0.</p><p>∴ Answer: <strong>0</strong> (or <strong>2nπ</strong> in general form)</p>
Correct Answer: C