If the lines 3-x 2 = 5 y-2 3 +1 =5-z and x+2 -1 = 1-3 y 7 = 4-z 2 are at right angles, then 7 -10 =

Step-by-step solution

Transform the given line equations into symmetric form to extract direction ratios. For the first line, 3-x 2 = 5y-2 3 +1 = 5-z , rewrite as: x-3 -2 = y - 2 5 3 +1 5 = z-5 -1 The direction ratios are d₁ = -2, 3 +1 5 , -1 . For the second line, x+2 -1 = 1-3y 7 = 4-z 2 , rewrite as: x+2 -1 = y - 1 3 - 7 3 = z-4 -2 The direction ratios are d₂ = -1, - 7 3 , -2 . Apply the perpendicularity condition d₁ d₂ = 0 : (-2)(-1) + ( 3 +1 5 ) (- 7 3 ) + (-1)(-2 ) = 0 2 - 7(3 +1) 15 + 2 = 0 Multiply the entire equation by 15 to el