Show that the lines
`vec(r )=(3hat(i) -15hat(j) + 9hat(k)) + lambda(2hat(i) -7hat(j) +5hat(k))` and
`vec(r )=(-hat(i) + hat(j) + 9hat(k)) + mu (2hat(i) + hat(j) -3hat(k))`
do not intersect.

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 .

vec(r )=(-4hat(i)+4hat(j) +hat(k)) + lambda (hat(i) +hat(j) -hat(k)) vec(r)=(-3hat(i) -8hat(j) -3hat(k)) + mu (2hat(i) +3hat(j) +3hat(k))

Find the vector and cartesian equations of a plane containing the two lines vec(r) = (2 hat(i) + hat(j) - 3 hat(k)) + lambda(hat(i) + 2 hat(j) + 5 hat(k)) and vec(r) = (3hat(i) + 3 hat(j) + 2 hat(k)) + lambda (3 hat(i) - 2 hat(j) + 5 hat(k)) . Also show that the line vec(r) = (2 hat(i) + 5 hat(j) + 2 hat(k)) + p (3 hat(i) - 2 hat(j) + 5 hat(k)) lies in the plane.

Find the shortest distance between the lines : vec(r) = (4hat(i) - hat(j)) + lambda(hat(i) + 2hat(j) - 3hat(k)) and vec(r) = (hat(i) - hat(j) + 2hat(k)) + mu (2hat(i) + 4hat(j) - 5hat(k))