<p>A line segment has direction cosines \(l, m, n\). If the line makes an angle of \(45°\) with the \(x\)-axis and \(120°\) with the \(y\)-axis, then the angle made by the line with the positive \(z\)-axis is:</p>
Step-by-Step Solution
Key Concept: Use the fundamental property that l² + m² + n² = 1, where l = cos(α), m = cos(β), n = cos(γ) are direction cosines for angles with x, y, z axes respectively.
Step 1: Identify given information. The line makes 45° with x-axis and 120° with y-axis. We need the angle with positive z-axis. Step 2: Use the direction cosine property. If angles are α = 45°, β = 120°, γ = angle with z-axis, then: l = cos(45°) = 1/√2 m = cos(120°) = -1/2 n = cos(γ) Step 3: Apply the constraint l^2 + m^2 + n^2 = 1: (1/√2)^2 + (-1/2)^2 + cos^2(γ) = 1 1/2 + 1/4 + cos^2(γ) = 1 3/4 + cos^2(γ) = 1 cos^2(γ) = 1/4 cos(γ) = ±1/2 Step 4: Since we need the angle with the positive z-axis, take the positive value: cos(γ) = 1/2, which gives γ = 60° ∴ Answer: C (60°)
Correct Answer: C