A line L passes through the point P(1, 2, 3) and is perpendicular to the lines L₁: x 1 = y -1 = z 0 and L₂: x 0 = y 1 = z -1 . Let M be the foot of the perpendicular drawn from the point Q(1, 5, 3) to the line L . A plane passes through M and is perpendicular to the line segment QM . The perpendicular distance of the point (2, 1, -1) from the plane is

Step-by-step solution

The direction ratios of the line L are given by the cross product of the direction vectors of L₁ and L₂ . b = vmatrix i & j & k 1 & -1 & 0 0 & 1 & -1 vmatrix = i (1 - 0) - j (-1 - 0) + k (1 - 0) = (1, 1, 1) The equation of the line L passing through P(1, 2, 3) is r = (1, 2, 3) + t(1, 1, 1) . Let M be the foot of the perpendicular from Q(1, 5, 3) to L . The coordinates of M are (1 + t, 2 + t, 3 + t) . The direction ratios of QM are (1 + t - 1, 2 + t - 5, 3 + t - 3) = (t, t - 3, t) . Since QM is perpendicular to L ,