Let $L _ { 1 }$ and $L _ { 2 }$ denote the lines
$$\begin{aligned}
& \vec { r } = \hat { i } + \lambda ( - \hat { i } + 2 \hat { j } + 2 \hat { k } ) , \lambda \in \mathbb { R } \text { and } \\
& \vec { r } = \mu ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) , \mu \in \mathbb { R }
\end{aligned}$$
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) $L _ { 3 }$?\\
(A) $\vec { r } = \frac { 2 } { 9 } ( 4 \hat { i } + \hat { j } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$\\
(B) $\vec { r } = \frac { 2 } { 9 } ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$\\
(C) $\vec { r } = \frac { 1 } { 3 } ( 2 \hat { i } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$\\
(D) $\vec { r } = t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$