<p>If \(|z_1 - 1| \leq 1\), \(|z_2 - 2| \leq 2\), \(|z_3 - 3| \leq 3\), then find the greatest value of \(|z_1 + z_2 + z_3|\).</p>
Step-by-Step Solution
Key Concept: Use the reverse triangle inequality: |z₁ + z₂ + z₃| ≤ |z₁| + |z₂| + |z₃|, then maximize each |zᵢ| by positioning zᵢ on the boundary of its constraint region along the positive real axis.
<p><strong>Step 1:</strong> Interpret the constraints. Each inequality describes a disk in the complex plane:</p><ul><li>|z₁ - 1| ≤ 1 ⟹ z₁ lies in disk centered at 1 with radius 1</li><li>|z₂ - 2| ≤ 2 ⟹ z₂ lies in disk centered at 2 with radius 2</li><li>|z₃ - 3| ≤ 3 ⟹ z₃ lies in disk centered at 3 with radius 3</li></ul><p><strong>Step 2:</strong> Find maximum |zᵢ| for each i. For a complex number in disk centered at center c with radius r:</p><p>max|zᵢ| = |center| + radius</p><ul><li>max|z₁| = |1| + 1 = 2</li><li>max|z₂| = |2| + 2 = 4</li><li>max|z₃| = |3| + 3 = 6</li></ul><p><strong>Step 3:</strong> Apply triangle inequality. By the triangle inequality:</p><p>|z₁ + z₂ + z₃| ≤ |z₁| + |z₂| + |z₃| ≤ 2 + 4 + 6 = 12</p><p><strong>Step 4:</strong> Verify equality is achievable. Equality holds when z₁, z₂, z₃ all point in the same direction (positive real axis):</p><ul><li>z₁ = 2 (on ray from 1 through 1, distance 1)</li><li>z₂ = 4 (on ray from 2 through 2, distance 2)</li><li>z₃ = 6 (on ray from 3 through 3, distance 3)</li></ul><p>Then z₁ + z₂ + z₃ = 2 + 4 + 6 = 12</p><p><strong>∴ Greatest value of |z₁ + z₂ + z₃| = 12</strong></p>
Correct Answer: 12