The lines x-1 2 = y-4 4 = z-2 3 and 1-x 1 = y-2 5 = 3-z a are perpendicular to each other, then a equals to

Step-by-step solution

The first line is given by x-1 2 = y-4 4 = z-2 3 . The direction vector of this line is v₁ = 2 i + 4 j + 3 k . The second line is given by 1-x 1 = y-2 5 = 3-z a . Rewriting this in standard form x-1 -1 = y-2 5 = z-3 -a , the direction vector is v₂ = -1 i + 5 j - a k . Since the lines are perpendicular, the dot product of their direction vectors must be zero, i.e., v₁ v₂ = 0 . (2)(-1) + (4)(5) + (3)(-a) = 0 -2 + 20 - 3a = 0 18 - 3a = 0 3a = 18 a = 6 Answer: 6