If `L_(1):(vec r)=(-hat i-2hat j-hat k)+lambda(3hat i+hat j+2hat k)` and `L_(2):vec r=(2hat i-2hat j+3hat k)+mu(hat i+2hat j+3hat k)`are two lines, then a vector perpendicular to both the lines of magnitude `5sqrt(3)` is

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

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)

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))

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)

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

A point on the line vec r=2hat i+3hat j+4hat k+t(hat i+hat j+hat k) is

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.

Vectors vec A=hat i+hat j-2hat k and vec B=3hat i+3hat j-6hat k are

Let L_(1):vec r=(hat i-hat j+2 hat k)+lambda(hat i-hat j+2 hat k),lambda in R L_(2):vec r=(hat j-hat k)+mu(3 hat i+hat j+p hat k),mu in R ,and L_(3):vec r=delta(l hat i+m hat j+n hat k),delta in R be three lines such that L_(1) is perpendicular to L_(2) and L_(3) is perpendicular to both L_(1) and L_(2) .Then,the point which lies on L_(3) is