Consider two lines `vec r=hat i+hat j+lambda_(1)(hat i-hat k)` and `vec r=hat j+hat k+lambda_(2)(hat i+hat j)` .If PQ is the shortest distance between these two lines then find PQ

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)

The lines vec r=(hat i+hat j+hat k)alpha+3hat k and vec r=(hat i-2hat j+hat k)beta+3hat k

Consider a line vec r=hat i+t(hat i+hat j+hat k) and a plane (vec r-(hat i+hat j))*(hat i-hat j-hat k)=1

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

Find angle between the vectors vec a=hat i+hat j-hat k and vec b=hat i-hat j+hat k

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)+u(2hat i+4hat j-5hat k)

If vec r=(hat i+2hat j+3hat k)+lambda(hat i-hat j+hat k) and vec r=(hat i+2hat j+3hat k)+mu(hat i+hat j-hat k) are bisector of two lines is

Prove that the plane vec r=(hat i+2hat j-hat k)=3 contains the line vec r=hat i+hat j+lambda(2hat i+hat j+4hat k)

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

State when the line vec(r) = vec(a) + lambda vec(b) is parallel to the plane vec(r).vec(n) = d. Show that the line vec(r) = hat(i) + hat(j) lambda (3 hat(i) - hat(j) + 2 hat(k)) is parallel to the plane vec(r).(2 hat(i) + hat(k) ) = 3. Also, find the distance between the line and the plane.