<p>A line \(AB\) in three-dimensional space makes angles \(45°\) and \(120°\) with the positive \(x\)-axis and the positive \(y\)-axis, respectively. If \(AB\) makes an acute angle \(\theta\) with the positive \(z\)-axis, then \(\theta\) equals</p>
Step-by-Step Solution
Key Concept: The direction cosines of a line satisfy l² + m² + n² = 1, where l, m, n are cosines of angles with x, y, z axes respectively. Use this constraint to find the angle with the z-axis.
Step 1: Let the line AB make angles α = 45°, β = 120°, and γ = θ with the positive x, y, and z axes respectively. Step 2: The direction cosines are l = cos(45°) = 1/√2, m = cos(120°) = -1/2, and n = cos(θ). Step 3: Apply the fundamental constraint for direction cosines: l^2 + m^2 + n^2 = 1 (1/√2)^2 + (-1/2)^2 + cos^2(θ) = 1 1/2 + 1/4 + cos^2(θ) = 1 Step 4: Solve for cos^2(θ): cos^2(θ) = 1 - 1/2 - 1/4 = 1/4 cos(θ) = ±1/2 Step 5: Since θ is acute, cos(θ) > 0, so cos(θ) = 1/2 Therefore, θ = 60° ∴ Answer: B
Correct Answer: B