The equation of the line, passing through $$(1,2,3)$$ and parallel to planes $$x-y+2 z=5$$ and $$3 x+y+z=6$$, is

The shortest distance (in units) between the lines $$\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}$$ and $$\bar{r}=(2 \hat{i}-2 \hat{j}+3 \hat{k})+\lambda(\hat{i}+2 \hat{j})$$ is

The length (in units) of the projection of the line segment, joining the points $$(5,-1,4)$$ and $$(4,-1,3)$$, on the plane $$x+y+z=7$$ is

If the volume of tetrahedron, whose vertices are $$\mathrm{A}(1,2,3), \mathrm{B}(-3,-1,1), \mathrm{C}(2,1,3)$$ and $$D(-1,2, x)$$ is $$\frac{11}{6}$$ cubic units, then the value of $$x$$ is