Let a point P on the line L₁ : x-2 1 = y+1 2 = z-1 1 be at a minimum distance from the point R(2, 2, 1) . If Q( , , ) is the mirror image of the point P in the line L₂ : x-1 3 = y-2 4 = z-3 5 , then the value of 3 + 4 + 5 is equal to

Step-by-step solution

Let the coordinates of point P on the line L₁ be ( +2, 2 -1, +1) . The vector RP is given by ( +2-2) i + (2 -1-2) j + ( +1-1) k = i + (2 -3) j + k . For P to be at a minimum distance from R , the vector RP must be perpendicular to the line L₁ . The direction ratios of L₁ are (1, 2, 1) . Using the condition of perpendicularity: 1( ) + 2(2 -3) + 1( ) = 0 + 4 - 6 + = 0 6 = 6 = 1 Thus, the coordinates of P are (3, 1, 2) . Now, Q( , , ) is the mirror image of P(3, 1, 2) in the line L₂ . The vector PQ is perpendicular to