Find the shortest distance between the lines `l_(1)and l_(1)` whose vector equations are
`vecr=(hati+hatj) + lambda (3hati + 4hatj - 2hatk) …(i)`
and `vecr=(2hati+3hatj) + mu (6hati + 8hatj - 4hatk) …(ii)`

Explanation

Comparing (i) and (ii) with
`vecr = veca + lambdavecb, veca_(1)=hati+hatj, veca_(2)=2hati+3hatj and vecb = 3hati + 4hatj-2hatk`
as lines `l_(1) and l_(2)` are parallwl to each other
Also we know that shortest distance between parallel lines is given by
`abs(((veca_(2)-veca_(1))xxvecb)/abs(vecb))`
so distance beteen (i) and (ii) is given by
`abs(((hati+2hatj)xx(3hati+4hatj-2hatk))/(sqrt(29)))`
here `(veca_(2)-veca_(1))xxvecb=(hati+2hatj)xx (3hati+4j-2hatk)`
`=abs[[hati,hatj,hatk],[1, 2, 0],[3,4,-2]]= -4hati + 2hatj - 2hatk`
`abs((hati+2hatj)xx(3hati+4hatj-2hatk))=2abs((-2hati+hatj-hatk))`
`=2sqrt(6)`
Distance between (i) and (ii)
`abs((2sqrt6)/(sqrt29))=2sqrt(6/29)`