Complex Numbers
Argument of complex numbers
Grade 11
Question:
<p>If \(|z_1| = \sqrt{2}\), \(|z_2| = \sqrt{3}\) and \(|z_1 + z_2| = \sqrt{5 - 2\sqrt{3}}\), then \(\arg\left(\dfrac{z_1}{z_2}\right)\) (not necessarily principal) is</p>
<p>\(\dfrac{3\pi}{4}\)</p>
<p>\(\dfrac{2\pi}{3}\)</p>
<p>\(\dfrac{5\pi}{4}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: Use the identity |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁·z̄₂) to find Re(z₁·z̄₂), then apply z₁·z̄₂ = |z₁||z₂|e^(i(arg(z₁)-arg(z₂))) to extract the argument difference.
<p><strong>Step 1:</strong> Expand |z₁ + z₂|² using the formula |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁·z̄₂).</p><p>5 - 2√3 = 2 + 3 + 2Re(z₁·z̄₂)</p><p>5 - 2√3 = 5 + 2Re(z₁·z̄₂)</p><p>Re(z₁·z̄₂) = -√3</p><p><strong>Step 2:</strong> Express z₁·z̄₂ in polar form. Let z₁·z̄₂ = |z₁||z₂|e^(iθ) where θ = arg(z₁) - arg(z₂).</p><p>z₁·z̄₂ = √2·√3·e^(iθ) = √6·e^(iθ)</p><p><strong>Step 3:</strong> Since Re(z₁·z̄₂) = -√3, we have:</p><p>√6·cos(θ) = -√3</p><p>cos(θ) = -√3/√6 = -1/√2 = -√2/2</p><p><strong>Step 4:</strong> The angles satisfying cos(θ) = -√2/2 are θ = 3π/4 + 2πn or θ = -3π/4 + 2πn.</p><p>Therefore, arg(z₁/z₂) = ±3π/4 + 2πn (where n ∈ ℤ)</p><p>∴ Answer: B</p>
Correct Answer: B