Complex Numbers
Modulus and Argument
Grade 11
Question:
<p>If \(|Z_1 + Z_2| = |Z_1| + |Z_2|\), then which of the following is correct?</p>
<p>\(Z_1 \bar{Z}_2\) is a positive real number</p>
<p>\(Z_1 \bar{Z}_2\) is a negative real number</p>
<p>\(\arg(Z_1) + \arg(Z_2) = 0\)</p>
<p>\(\arg(Z_1) - \arg(Z_2) = \pi\)</p>
Step-by-Step Solution
Key Concept: The triangle inequality |Z₁ + Z₂| ≤ |Z₁| + |Z₂| becomes an equality if and only if Z₁ and Z₂ have the same argument (point in the same direction), meaning Z₂ = kZ₁ where k ≥ 0 is real.
<p><strong>Step 1:</strong> Recall the triangle inequality for complex numbers: |Z₁ + Z₂| ≤ |Z₁| + |Z₂|</p><p><strong>Step 2:</strong> Equality |Z₁ + Z₂| = |Z₁| + |Z₂| holds if and only if Z₁ and Z₂ point in the same direction, i.e., arg(Z₁) = arg(Z₂)</p><p><strong>Step 3:</strong> This occurs when Z₂ = kZ₁ for some real k ≥ 0 (or equivalently, Z₁ = mZ₂ for some real m ≥ 0)</p><p><strong>Step 4:</strong> Algebraically: Z₁ and Z₂ have the same argument ⟺ Z₂/Z₁ is a positive real number ⟺ arg(Z₂) - arg(Z₁) = 0 (mod 2π)</p><p>∴ The correct answer is: <strong>Z₂ = kZ₁ where k is a positive real number</strong> (or Z₁ and Z₂ have the same argument)</p>
Correct Answer: A