If A (1, 2, 3), B (3, 1, 2), C(0, 1, 1) and D(2, 0, 4) are four points then find the shortest distance between the line AB and CD.

Three lines are given by

$\overrightarrow{r}=\lambda \hat{i}, \lambda \in R$,

$\overrightarrow{r}=\mu \left(\widehat{i}+\widehat{j}\right),\mu \in R$ and

$\overrightarrow{r}=\nu \left(\widehat{i}+\widehat{j}+\widehat{k}\right),\nu \in R.$

Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $ABC$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

Lines $\overrightarrow{r}=\left(3+t\right)\hat{i}+\left(1-t\right)\hat{j}+\left(-2-2t\right)\hat{k}, t\in R$ and $x=4+k, y=-k, z=-4-2k, k\in R,$ then relation between lines is