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JEE Main20266 April 2026Morning ShiftMathematicsThree Dimensional GeometryActual

Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 = z 2 , then is equal to:

Options

  1. A3 2 2
  2. B5 2 2
  3. C5 3 2
  4. D4 3 2

Correct answer

B. 5 2 2

Step-by-step solution

The direction vector of line L₁ is d₁ = 3 i + 5 j + 7 k . The direction vector of line L₂ is d₂ = i + 4 j + 7 k . Since line L is perpendicular to both L₁ and L₂ , its direction vector d is given by the cross product of d₁ and d₂ : d = d₁ d₂ = vmatrix i & j & k 3 & 5 & 7 1 & 4 & 7 vmatrix d = i (35 - 28) - j (21 - 7) + k (12 - 5) = 7 i - 14 j + 7 k The direction ratios of L are proportional to 1, -2, 1 . The direction vector of line L₃ is d₃ = 2 i + j + 2 k . The acute angle between L and L₃ is given by: = | d d₃ |

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