Answer
$\sqrt{202}$
Work Step by Step
The first job is to find pairs of parallel vectors, so we are able to tell which two vectors that originate from a point (say, A) define the parallelogram.
The three vectors with initial point A are
$ \overrightarrow{AB} = \langle 1-1, 7-0,2+1 \rangle= \langle 0, 7, 3 \rangle$
$ \overrightarrow{AC} = \langle 1, 4, 0 \rangle$
$ \overrightarrow{AD} = \langle-1, 3, 3 \rangle$
We search for parallel vectors
$ \overrightarrow{BC} = \langle 1, -3, -3 \rangle,$ which is parallel to $ \overrightarrow{AD}$,
$ \overrightarrow{BD} = \langle-1, -4, 0 \rangle,$ which is parallel to $ \overrightarrow{AC}$,
We have our pairs of parallel vectors.
The vectors defining the paralelogram are $\overrightarrow{AC}$ and $\overrightarrow{AD}$ ($\overrightarrow{AB}$ is the diagonal).
Area =$| \overrightarrow{AC}\times \overrightarrow{AD} |$
$\overrightarrow{AC}\times\overrightarrow{AD}$=$\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
1 & 4 & 0\\
-1 & 3 & 3
\end{array}\right|=\langle (12-0) ,\ -(3-0) ,\ (3+4) \rangle$
$=\langle 12 ,\ -3 ,\ 7 \rangle$
Area =$| \overrightarrow{AB}\times\overrightarrow{AD} |$=$\sqrt{144+9+49}=\sqrt{202}$
