Let a line L₁ pass through the origin and be perpendicular to the lines L₂: r = (3+t) i + (2t-1) j + (2t+4) k and L₃: r = (3+2s) i + (3+2s) j + (2+s) k , t, s R . If (a, b, c) , a Z , is the point on L₃ at a distance of 17 from the point of intersection of L₁ and L₂ , then (a+b+c)^2 is equal to ________.

Step-by-step solution

The direction vectors of the given lines L₂ and L₃ are d₂ = i + 2 j + 2 k and d₃ = 2 i + 2 j + k respectively. Since line L₁ is perpendicular to both L₂ and L₃ , its direction vector d₁ is given by the cross product of d₂ and d₃ : d₁ = d₂ d₃ = vmatrix i & j & k 1 & 2 & 2 2 & 2 & 1 vmatrix = -2 i + 3 j - 2 k Since L₁ passes through the origin, its equation is: r = (-2 i + 3 j - 2 k ) Let the point of intersection of L₁ and L₂ be P . The coordinates of P can be written as (-2 , 3 , -2 ) from L₁ and (3+t, 2t-1, 2t+4)