Let l₁ be the line joining (1, 1, 1) and (3, 1, 3) and let l₂ be the line joining (0, 2, -1) and (2, 0, 3) . What is the angle between l₁ and l₂ ?

Step-by-step solution

Direction ratios of line l₁ joining (1, 1, 1) and (3, 1, 3) are (3-1, 1-1, 3-1) = (2, 0, 2) . Direction ratios of line l₂ joining (0, 2, -1) and (2, 0, 3) are (2-0, 0-2, 3-(-1)) = (2, -2, 4) . Let be the angle between l₁ and l₂ . Using the formula for the angle between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) : = |a₁ a₂ + b₁ b₂ + c₁ c₂| a₁^2 + b₁^2 + c₁^2 a₂^2 + b₂^2 + c₂^2 Substituting the values: = |(2)(2) + (0)(-2) + (2)(4)| 2^2 + 0^2 + 2^2 2^2 + (-2)^2 + 4^2 = |4 + 0 + 8| 8 24 = 12 2 2 2 6