Question 6
In the three-dimensional Cartesian space, let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote unit vectors along three mutually perpendicular directions $x$, $y$ and $z$-axes respectively. Three straight lines $l_1$, $l_2$ and $l_3$ have equations defined by
$$\begin{aligned}l_1:\ \mathbf r&=(4+\lambda)\mathbf i+(5+\lambda)\mathbf j+(6+\lambda)\mathbf k,\\l_2:\ \mathbf r&=(4+3\mu)\mathbf i+(5-\mu)\mathbf j+(6-2\mu)\mathbf k,\\l_3:\ \mathbf r&=(1+6\nu)\mathbf i+(2+2\nu)\mathbf j+(3+\nu)\mathbf k,\end{aligned}$$
where $\mu$, $\lambda$ and $\nu$ are real numbers. If the area of the triangle enclosed by the three lines $l_1$, $l_2$ and $l_3$ is denoted by $S$, find the value of $10S^2$.
$$\begin{aligned}l_1:\ \mathbf r&=(4+\lambda)\mathbf i+(5+\lambda)\mathbf j+(6+\lambda)\mathbf k,\\l_2:\ \mathbf r&=(4+3\mu)\mathbf i+(5-\mu)\mathbf j+(6-2\mu)\mathbf k,\\l_3:\ \mathbf r&=(1+6\nu)\mathbf i+(2+2\nu)\mathbf j+(3+\nu)\mathbf k,\end{aligned}$$
where $\mu$, $\lambda$ and $\nu$ are real numbers. If the area of the triangle enclosed by the three lines $l_1$, $l_2$ and $l_3$ is denoted by $S$, find the value of $10S^2$.