Question

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0

Answer 1

Given equations of lines are x = y, z = 0

i.e., `x/1 = "y"/1 = "z"/0` and x = 0,

z = 0 which represents Y-axis.

Direction ratios of above lines are

a₁ = 1, b₁ = 1, c₁ = 0 and

a₂ = 0, b₂ = 1, c₂ = 0   ......(Direction ratios of Y-axis)

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)(0) + (1)(1) + (0)(0))/(sqrt(1^2 + 1^2 + 0^2), sqrt(0^2 + 1^2 + 0^2))|`

∴ cos θ = `|1/(sqrt(2), sqrt(1))|`

∴ θ = `cos^-1(1/sqrt(2))`

∴ θ = 45°