3D Geometry
Direction Cosines and Direction Ratios of a Line
Grade 12
Question:
<p>If a line makes an angle of \(\pi/4\) with the positive directions of each of <em>x</em>-axis and <em>y</em>-axis, then the angle that the line makes with the positive direction of the <em>z</em>-axis is</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the property that the sum of squares of direction cosines equals 1: l² + m² + n² = 1, where l, m, n are cosines of angles with x, y, z axes respectively.
Step 1: Let the line make angles α, β, γ with the positive x, y, z axes respectively. Given: α = π/4 and β = π/4 Step 2: The direction cosines are l = cos(π/4) = 1/√2, m = cos(π/4) = 1/√2, n = cos(γ) Step 3: Apply the fundamental identity: l^2 + m^2 + n^2 = 1 (1/√2)^2 + (1/√2)^2 + cos^2(γ) = 1 1/2 + 1/2 + cos^2(γ) = 1 Step 4: Solve for cos^2(γ): cos^2(γ) = 0 cos(γ) = 0 Step 5: Therefore γ = π/2 ∴ Answer: D (The angle with z-axis is π/2)
Correct Answer: D