Let the lines L₁ and L₂ be given by L₁: x-1 2 = y-3 1 = z-1 -2 and L₂: x+2 1 = y 2 = z-1 2 intersect at a point P . A third line L₃ passes through P and is perpendicular to both L₁ and L₂ . Which of the following points lies on L₃ and is at

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Let the lines L₁ and L₂ be given by L₁: x-1 2 = y-3 1 = z-1 -2 and L₂: x+2 1 = y 2 = z-1 2 intersect at a point P . A third line L₃ passes through P and is perpendicular to both L₁ and L₂ . Which of the following points lies on L₃ and is at a distance of 6 units from P ?

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Correct answer: B. (3, -2, 5)

Let the coordinates of a point on L₁ be (1+2 , 3+ , 1-2 ) and on L₂ be (-2+ , 2 , 1+2 ) . For the point of intersection P , we equate the respective coordinates: 1+2 = -2+ - 2 = 3 3+ = 2 2 - = 3 1-2 = 1+2 = - Solving these equations, we get = -1 and = 1 . Substituting = -1 into the parametric coordinates of L₁ , the point of intersection is P(-1, 2, 3) . The direction vectors of L₁ and L₂ are d ₁ = 2 i + j - 2 k and d ₂ = i + 2 j + 2 k . The direction vector of L₃ is perpendicular to both d ₁ and d ₂ , so it is par

Let the lines L₁ and L₂ be given by L₁: x-1 2 = y-3 1 = z-1 -2 and L₂: x+2 1 = y 2 = z-1 2 intersect at a point P . A third line L₃ passes through P and is perpendicular to both L₁ and L₂ . Which of the following points lies on L₃ and is at a distance of 6 units from P ?

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