Question

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

Answer 1

Given equations of lines are `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

Direction ratios of above lines are

a₁ = 1, b₁ = −1, c₁ = 2 and a₂ = 2, b₂ = 1, c₂ = 1

Angle between two lines is

cos θ = `|("a"_1"a"_2 + "b"_1"b"_2 + "c"_1"c"_2)/(sqrt("a"_1^2 + "b"_1^2 + "c"_1^2) sqrt("a"_2^2 + "b"_2^2 + "c"_2^2))|`

∴ cos θ = `|((1)(2) + (-1)(1) + (2)(1))/(sqrt(1^2 + (-1)2 + 2^2 sqrt(2^2 + 1^2 + 1^2))|`

∴ cos θ = `|(2 - 1 + 2)/(sqrt(6)sqrt(6))|`

∴ cos θ = `|3/6|`

∴ θ = `cos^-1(1/2)`

∴ θ = 60°