Show that the lines
`vec(r) =(hat(i) +2hat(j) +hat(k)) +lambda (hat(i)-hat(j)+hat(k)) " and " vec(r ) =(hat(i) +hat(j) -hat(k)) + mu (hat(i)- hat(j) + 2hat(k))`
Do not intersect .

Explanation

The given lines are
`vec(r ) =(hat(i) +2hat(j) +hat(k)) +lambda (hat(i) -hat(j) +hat(k))`
` vec(r) =(hat(i) +hat(j)+hat(k)) +mu (hat(i)-hat(j)+hat(k))`
These lines will intesect if for some particular values of `lambda " and " mu ` the values of `vec(r ) ` given by (1) and (2) are the same.
This happens when
`(hat(i) +2hat(j) + hat(k)) +lambda (hat(i)-hat(j) +hat(k)) =(hat(i) +hat(j) +hat(k))+mu (hat(i)-ha(j) +2hat(k))`
`rArr (1+lambda)hat(i) +(2-lambda)hat(j)+(1+lambda)hat(k)=(1+mu)hat(i) +(1-mu)hat(j)+(1+2mu)hat(k)`
`rArr 1+lambda =1+mu , 2-lambda =1 -mu " and " 1+lambda =1+2mu`
`rArr lambda-mu=0 ....(i) , lambda-mu=1 .....(ii) , lambda-2mu=0 .....(iii)`
From (ii) and (iii) we get `lambda=2 " and " mu =1 `
And these values of `lambda "and " mu` do not satisfy (i)
Hence the given lines do not intersect .