Question

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

Answer 1

Let `bar"a" and bar"b"` be the vectors in the direction of the lines `(x - 1)/(1) = (y - 2)/(-1) = (z - 3)/(2) and (x - 1)/(2) = (y - 2)/(1) = (z - 3)/(1)` respectively.

Then `bar"a" = hat"i" - hat"j" + 2hat"k", bar"b" = 2hat"i" + hat"j" + hat"k"`

∴ `bar"a".bar"b" = (hat"i" - hat"j" + 2hat"k").(2hat"i" + hat"j" + hat"k")`

= (1)(2) + (– 1)(1) + (2)(1)
= 2 – 1 + 2
= 3
`|bar"a"| = sqrt(1^2 + (-1)^2 + 2^2) = sqrt(6)`

`|bar"b"| = sqrt(2^2 + 1^2 + 1^2) = sqrt(6)`

If θ is the angle between the lines, then

cos θ = `(bar"a".bar"b")/(|bar"a"||bar"b"|) =(3)/(sqrt(6)sqrt(6)) = (1)/(2)` = cos 60°

∴ θ = 60°.