Question

Find the acute angle between the lines `(x - 1)/1 = (y - 2)/-1 = (z - 3)/2` and `barr = (hati + 2hatj + 3hatk) + λ(2hati + hatj + hatk)`.

Answer 1

Vectors along the given lines are

`bara = hati - hatj + 2hatk` and `barb = 2hati + hatj + hatk`

Angle between `bara` and `barb` is the angle between the lines.

∴ The acute angle θ between the lines is given by

cos θ = `|(bara*barb)/(|bara||barb|)|`

= `|((hati - hatj + 2hatk)*(2hati + hatj + hatk))/(sqrt(1 + 1 + 4)*sqrt(4 + 1 + 1))|`

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

= `|(2 - 1 + 2)/6|`

= `3/6`

= `1/2`

∴ θ =  `cos^-1  1/2` = 60°