Let $L_1$ and $L_2$ denote the lines
$\vec{r}=\hat{i}+\lambda\left(-\hat{i}+2\hat{j}+2\hat{k}\right), \lambda\,\in\,\mathbb{R}$ and
$\vec{r}=\mu\left(2\hat{i}-\hat{j}+2\hat{k}\right), \mu\,\in\,\mathbb{R}$
respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?