3D Geometry
Direction Cosines and Angles
Grade 12
Question:
<p>A line in the 3-dimensional space makes an angle \(\theta\,(0 < \theta \le \pi/2)\) with both the \(x\) and \(y\) axis. Then the set of all values of \(\theta\) is the interval</p>
<p>\(\left(0, \dfrac{\pi}{4}\right]\)</p>
<p>\(\left[\dfrac{\pi}{6}, \dfrac{\pi}{3}\right]\)</p>
<p>\(\left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\right]\)</p>
<p>\(\left(\dfrac{\pi}{3}, \dfrac{\pi}{2}\right]\)</p>
Step-by-Step Solution
Key Concept: The direction cosines of a line satisfy l² + m² + n² = 1, and the angle with a coordinate axis determines a specific direction cosine value. Use the constraint equation to find the range of possible angles with the other axes.
Step 1: If a line makes angle θ with the x-axis, then cos θ = l where l is the direction cosine. Since 0 < θ < π/2, we have 0 < l < 1. Step 2: Using the constraint l^2 + m^2 + n^2 = 1, we get m^2 + n^2 = 1 - l^2 = 1 - cos^2θ = sin^2θ. Step 3: For the angle α with the y-axis: cos^2α = m^2 ≤ m^2 + n^2 = sin^2θ, so |cos α| ≤ sin θ, which means α ≥ arccos(sin θ). Step 4: Since sin θ = cos(π/2 - θ), the minimum value of α occurs when cos α = sin θ = cos(π/2 - θ), giving α_min = π/2 - θ. Step 5: As m^2 varies from 0 to sin^2θ, α varies from π/2 to (π/2 - θ). Therefore, the line makes an angle in the range [π/2 - θ, π/2] with the y-axis (and similarly with the z-axis). ∴ Answer: C
Correct Answer: C