The measure of the angle between the lines x=k+1, y=2 k-1, z=2 k+3, k R and x-1 2 = y+1 1 = z-1 -2 is

Step-by-step solution

The first line is given by x = k+1 , y = 2k-1 , z = 2k+3 . This can be written in symmetric form as x-1 1 = y+1 2 = z-3 2 = k . The direction vector of this line is v₁ = i + 2 j + 2 k . The second line is given by x-1 2 = y+1 1 = z-1 -2 . The direction vector of this line is v₂ = 2 i + j - 2 k . The angle between the two lines is given by = | v₁ v₂ | | v₁ | | v₂ | . Calculating the dot product: v₁ v₂ = (1)(2) + (2)(1) + (2)(-2) = 2 + 2 - 4 = 0 . Since the dot product is 0 , the lines are perpendicular to each other