Show that the lines ` vec r=( hat i+ hat j- hat k)+lambda(3 hat i- hat j)a n d vec r=(4 hat i- hat k)+mu(2 hat i+3 hat k)` are coplanar. Also, find the plane containing these two lines.

Explanation

The given lines are
`vec(r ) =(hat(i) +hat(j) -hat(k)) +lambda (3hat(i) -hat(j))`
`vec( r) =(4hat(i) -hat(k)) +mu (2hat(i) +3hat(k))`
These lines will intersect if for some particular values of `lambda " and " mu` the values `vec( r) ` given by (1) and (2) are the same
So, we must have
`(hat(i) +hat(j)-hat(k))+lambda(3hat(i)-hat(j))=(4hat(i)-hat(k))+mu(2hat(i)+3hat(k))`
`rArr (1+3lambda)hat(i) +(1-lambda)hat(j) -hat(k)=(4+2mu) hat(i) +(3mu-1)hat(k)`
`rArr 1+3lambda =4+2mu,1 -lambda =0 " and " 3mu-1 =-1`
`rArr 3lambda -2mu =3 .....(i),lambda , =1 ...(i) " and " mu =0 .....(iii)`
Clearly `lambda=1 " and " mu =0` also satisfy (i)
Hence the given lines intersect .
Putting `lambda =1 " in (1) we get " vec(r ) =(4hat(i) +0hat(j) -hat(k))`
Hence the point of intersection of the given lines is (4,0 ,-1 )