Complex Numbers
Argument of complex numbers
Grade None
Question:
<p><strong>For Problems 5–8</strong><br>Consider the complex numbers \(z_1\) and \(z_2\) satisfying the relation \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2\).</p><p><strong>Problem 8.</strong> Possible difference between the argument of \(z_1\) and \(z_2\) is</p>
<p>(1) \(0\)</p>
<p>(2) \(\pi\)</p>
<p>(3) \(-\dfrac{\pi}{2}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: The condition |z₁ + z₂|² = |z₁|² + |z₂|² means that z₁ and z₂ are orthogonal in the complex plane (their dot product is zero), which occurs when arg(z₁) - arg(z₂) = ±π/2.
<p><strong>Step 1:</strong> Expand |z₁ + z₂|²:</p><p>|z₁ + z₂|² = (z₁ + z₂)(z̄₁ + z̄₂) = |z₁|² + |z₂|² + z₁z̄₂ + z̄₁z₂</p><p><strong>Step 2:</strong> Note that z₁z̄₂ + z̄₁z₂ = 2Re(z₁z̄₂). Given condition gives:</p><p>|z₁|² + |z₂|² + 2Re(z₁z̄₂) = |z₁|² + |z₂|²</p><p>Therefore: Re(z₁z̄₂) = 0</p><p><strong>Step 3:</strong> Write z₁ = r₁e^(iθ₁) and z₂ = r₂e^(iθ₂). Then:</p><p>z₁z̄₂ = r₁r₂e^(i(θ₁-θ₂))</p><p>Re(z₁z̄₂) = r₁r₂cos(θ₁ - θ₂) = 0</p><p><strong>Step 4:</strong> This means cos(θ₁ - θ₂) = 0, so:</p><p>θ₁ - θ₂ = ±π/2 (or ±90°)</p><p>∴ The possible difference between arg(z₁) and arg(z₂) is <strong>±π/2</strong> or <strong>±90°</strong></p>
Correct Answer: C