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

Explanation

Let `bar a and barb` be the vector in the direction of lines `(x-1)/4`
` = (y-3)/1 = z/8 and (x-2)/2 = (y+1)/2`
` = (z-4)/1` respectively .
` :. bar a = 4 hati + hatj + 8 hat k`
and ` bar b = 2 hat I + 2 hatj + hatk`
Then, `bara.barb = 8 + 2 + 8 = 18`
` |abara| = sqrt(4^(2) + 1^(2) + 8^(2))`
` = sqrt(16 + 1 + 64) = sqrt(81) = 9`
` |barb| = sqrt(2^(2) + 2^(2) + 1^(2)) = sqrt(4 + 4 + 1) `
` = sqrt9 = 3` ltbr. If `theta` is the acute angle between two given lines, then
`cos theta = (bara*barb)/(|bara|*|barb|) = 18/(9 xx 3) = 2/3 `
` :. " "theta = cos^(-1) (2/3).`