<p>The angle between the lines whose direction cosines satisfy the equations \(l + m + n = 0\) and \(l^2 = m^2 + n^2\) is</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Use the two constraint equations to find direction cosines of both lines, then apply the dot product formula for angle between lines. The constraint l² = m² + n² combined with l² + m² + n² = 1 determines specific direction ratios.
Step 1: From the constraints, we have: <ul><li>l + m + n = 0 ... (1)</li><li>l^2 = m^2 + n^2 ... (2)</li><li>l^2 + m^2 + n^2 = 1 ... (3) [fundamental property]</li></ul> Step 2: From (2) and (3): l^2 + l^2 = 1, so l^2 = 1/2, giving l = ±1/√2 Step 3: From (1): m + n = ∓1/√2 From (2): m^2 + n^2 = 1/2 Step 4: Using (m + n)^2 = m^2 + n^2 + 2mn:
1/2 = 1/2 + 2mn, so mn = 0 Therefore either m = 0 or n = 0 Step 5: For l = 1/√2, m = 0, n = -1/√2: Direction cosines are (1/√2, 0, -1/√2)
For l = 1/√2, m = -1/√2, n = 0: Direction cosines are (1/√2, -1/√2, 0) Step 6: The dot product of direction cosines:
d_1·d_2 = (1/√2)(1/√2) + (0)(-1/√2) + (-1/√2)(0) = 1/2 Step 7: cos θ = |1/2| = 1/2
∴ θ = 60° or π/3 ∴ Answer: C
Correct Answer: C