A line L passes through the point A(1, 1, 1) and is perpendicular to both the lines L₁ : x-2 1 = y-3 0 = z-4 -1 and L₂ : x+1 0 = y+2 1 = z+3 -1 . Let Q and R be two points on the line L at a distance of 17 from the point P(2, 4, 6) . Then the square of the area of the triangle PQR is ________.

Step-by-step solution

The direction vectors of lines L₁ and L₂ are d₁ = i - k and d₂ = j - k . The direction vector d of line L is perpendicular to both d₁ and d₂ , so it is given by their cross product: d = d₁ d₂ = i (0 - (-1)) - j (-1 - 0) + k (1 - 0) = i + j + k The equation of line L passing through A(1, 1, 1) with direction ratios (1, 1, 1) is: x-1 1 = y-1 1 = z-1 1 = t Let T be the foot of the perpendicular from P(2, 4, 6) to line L . The coordinates of T are (t+1, t+1, t+1) . The direction ratios of PT are (t+1-2, t+1-4, t+1-6) =