If the line 1-x -3 =y= z+2 2 is perpendicular to the line 3 x-1 2 b =3-y= z-1 a , then find the value of 3 a+3 b

Step-by-step solution

The first line is given by x-1 3 = y 1 = z+2 2 . The direction vector of this line is v₁ = 3 i + 1 j + 2 k . The second line is given by 3x-1 2b = y-3 -1 = z-1 a . Rewriting this in standard form x - 1/3 2b/3 = y-3 -1 = z-1 a . The direction vector of this line is v₂ = 2b 3 i - 1 j + a k . Since the lines are perpendicular, the dot product of their direction vectors must be zero: v₁ v₂ = 0 . (3) ( 2b 3 ) + (1)(-1) + (2)(a) = 0 . 2b - 1 + 2a = 0 . 2a + 2b = 1 . We need to find the value of 3a + 3b . 3(a + b) = 3 1 2