The angle between the lines, whose direction cosines $$l, \mathrm{~m}, \mathrm{n}$$ satisfy the equations $$l+\mathrm{m}+\mathrm{n}=0$$ and $$2 l^2+2 \mathrm{~m}^2-\mathrm{n}^2=0$$, is

Equation of the plane passing through $$(1,-1,2)$$ and perpendicular to the planes $$x+2 y-2 z=4$$ and $$3 x+2 y+z=6$$ is

A line with positive direction cosines passes through the point $$\mathrm{P}(2,-1,2)$$ and makes equal angles with the co-ordinate axes. The line meets the plane $$2 x+y+z=9$$ at point $$\mathrm{Q}$$. The length of the line segment $$P Q$$ equals

If the shortest distance between the lines $$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$$ and $$\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$$ is $$\frac{1}{\sqrt{3}}$$, then sum of possible values of $$\lambda$$ is