Let the lines L₁: x-1 2 = y+1 3 = z-2 4 and L₂: x-3 1 = y-k 2 = z 1 intersect at a point. If is the plane containing both the lines L₁ and L₂ , then the square of the perpendicular distance from the point (k, k-7, 4-2k) to the plane is ____

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Let the lines L₁: x-1 2 = y+1 3 = z-2 4 and L₂: x-3 1 = y-k 2 = z 1 intersect at a point. If is the plane containing both the lines L₁ and L₂ , then the square of the perpendicular distance from the point (k, k-7, 4-2k) to the plane is ____.

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Correct answer: 30

Any point on L₁ can be written as (2 +1, 3 -1, 4 +2) . Any point on L₂ can be written as ( +3, 2 +k, ) . Since the lines intersect, these points must coincide for some , : 2 + 1 = + 3 = 2 - 2 3 - 1 = 2 + k 4 + 2 = From the first and third equations: 4 + 2 = 2 - 2 2 = -4 = -2 Then, = 4(-2) + 2 = -6 . Substitute and into the second equation: 3(-2) - 1 = 2(-6) + k -7 = -12 + k k = 5 The direction vectors of L₁ and L₂ are d₁ = 2 i + 3 j + 4 k and d₂ = i + 2 j + k . The normal vector to the plane is: n = d₁ d₂ = vmatrix

Let the lines L₁: x-1 2 = y+1 3 = z-2 4 and L₂: x-3 1 = y-k 2 = z 1 intersect at a point. If is the plane containing both the lines L₁ and L₂ , then the square of the perpendicular distance from the point (k, k-7, 4-2k) to the plane is ____.

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