3D Geometry
Direction cosines
Grade 12
Question:
<p>If direction cosines of line \(L\) be \(l, m, n\), and \(2l + 3m + n = 0\) and \(l + 3m + 2n = 0\), then what is \(\cos\alpha\) where \(\alpha\) is the angle the line makes with the x-axis?</p>
<p>\(\dfrac{1}{\sqrt{2}}\)</p>
<p>\(\dfrac{1}{\sqrt{3}}\)</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(\dfrac{1}{\sqrt{5}}\)</p>
Step-by-Step Solution
Key Concept: Direction cosines satisfy l² + m² + n² = 1. Use the two constraint equations to express direction cosines in terms of a parameter, then normalize using this identity.
Step 1: Solve the system of constraints to find l:m:n. Given: 2l + 3m + n = 0 ... (1) and l + 3m + 2n = 0 ... (2) From (1) - (2): l - n = 0 ⟹ l = n Substituting in (2): l + 3m + 2l = 0 ⟹ 3l + 3m = 0 ⟹ m = -l Therefore: l : m : n = 1 : (-1) : 1 Step 2: Normalize using l^2 + m^2 + n^2 = 1. Let l = k, m = -k, n = k Then: k^2 + k^2 + k^2 = 1 ⟹ 3k^2 = 1 ⟹ k = ±1/√3 So the direction cosines are: l = ±1/√3, m = ∓1/√3, n = ±1/√3 Step 3: Find cos α where α is the angle with x-axis. The angle a line makes with the x-axis has cosine equal to l (the x-direction cosine). ∴ cos α = ±1/√3 = ±√3/3 Answer: B (Most likely |cos α| = 1/√3 or √3/3)
Correct Answer: B