Question

The angle between the lines `("x - 1")/4 = ("y - 3")/1 = "z"/8` and `("x - 2")/2 = ("y + 1")/2 = "z - 4"/1` is ______.

Options

• `cos^-1 (2/3)`

• `cos^-1 (1/2)`

• `cos^-1 (3/4)`

• `cos^-1 (1/3)`

Answer 1

The angle between the lines `("x - 1")/4 = ("y - 3")/1 = "z"/8` and `("x - 2")/2 = ("y + 1")/2 = "z - 4"/1` is `underline(cos^-1 (2/3))`.

Explanation:

Given line,

`("x - 1")/4 = ("y - 3")/1 = "z"/8` and 

`("x - 2")/2 = ("y + 1")/2 = "z - 4"/1`

Here, a₁ = 4, b₁ = 1, c₁ = 8

a₂ = 2, b₂ = 2, c₂ = 1

`therefore cos theta = ("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 theta = (8 + 2 + 8)/((sqrt(16 + 1 + 64))(sqrt(4 + 4 + 1)))`

`=> cos theta = 1/(9 xx 3) = 2/3`

`=> theta = cos^-1 (2/3)`