The angle between the lines x+4 3 = y-1 5 = z+3 4 and x+1 1 = y-4 1 = z-5 2 is

Step-by-step solution

The direction ratios of the first line are a₁ = 3 , b₁ = 5 , c₁ = 4 . The direction ratios of the second line are a₂ = 1 , b₂ = 1 , c₂ = 2 . The angle between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) is given by = |a₁ a₂ + b₁ b₂ + c₁ c₂| a₁^2 + b₁^2 + c₁^2 a₂^2 + b₂^2 + c₂^2 . Calculating the dot product: a₁ a₂ + b₁ b₂ + c₁ c₂ = (3)(1) + (5)(1) + (4)(2) = 3 + 5 + 8 = 16 . Calculating the magnitudes: a₁^2 + b₁^2 + c₁^2 = 3^2 + 5^2 + 4^2 = 9 + 25 + 16 = 50 = 5 2 . Calculating the magnitudes: a₂^2