A line L is passing through points A(1,3,2) and B(2,2,1) . If mirror image of point P (1,1,-1) in the line L is ( x , y , z ) then x+ y + z =

Step-by-step solution

Direction vector of the line through A(1,3,2) and B(2,2,1) is d = AB = (2-1, 2-3, 1-2) = (1,-1,-1) Parametric coordinates of any point on the line are M = (1+t, 3-t, 2-t) For the perpendicular foot from P(1,1,-1) , vector PM = (t, 2-t, 3-t) must satisfy PM d = 0 : t(1) + (2-t)(-1) + (3-t)(-1) = 0 t - 2 + t - 3 + t = 0 3t = 5 t = 5 3 Substituting gives the foot coordinates M = (1+ 5 3 , 3- 5 3 , 2- 5 3 ) = ( 8 3 , 4 3 , 1 3 ) Since M is the midpoint of P and its mirror image P'(x,y,z) , we solve: 1+x 2 = 8 3 x = 13