Question

Find the angle between the lines `(x -1)/4 = (y - 3)/1 = z/8`  and `(x-2)/2 = (y + 1)/2 = (z-4)/1`

Answer 1

Let `bara` and `barb` be the vectors in the direction of the lines `(x -1)/4 = (y - 3)/1 = z/8`  and `(x-2)/2 = (y + 1)/2 = (z-4)/1`respectively.

`:.bara = 4hati + hatj + 8hatk` and `barb = 2hati + 2hatj + hatk`

`:. bara.barb = 4 xx 2+1xx2+8xx1 = 8 + 2 + 8 = 18`

and

`|bara| = sqrt(16+1+64) = sqrt81 = 9`

`barb = sqrt(4+1+4) = sqrt9 = 3` 

Let Θ be the acute angle between the two given lines.

`:. costheta = |(bara.barb)/(|bara|.|barb|) = 18/(9xx3) = 2/3`

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