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

Step-by-step solution

The equation of the first line is 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 ratios of the first line are 1, 2, 2 . The equation of the second line is x - 1 2 = y - 2 1 = z - 3 1 . The direction ratios of the second line are 2, 1, 1 . Let be the angle between the two lines. Then, = |a₁ a₂ + b₁ b₂ + c₁ c₂| a₁^2 + b₁^2 + c₁^2 a₂^2 + b₂^2 + c₂^2 = |(1)(2) + (2)(1) + (2)(1)| 1^2 + 2^2 + 2^2 2^2 + 1^2 + 1^2 = 6 9 6 = 6 3 6 = 2 6 = 2 3 = ⁻¹