Find the equation of a plane passing through the points $A\left(2,1,2\right)$ and $B\left(4,-2,1\right)$ and perpendicular to plane $\overrightarrow{r}·\left(\hat{i}-2\hat{k}\right)=5$. Also, find the coordinates of the point, where the line passing through the points $\left(3,4,1\right)$ and $\left(5,1,6\right)$ crosses the plane thus obtained.

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