Answer
The volume of the parallelepiped: $V = 4$
Work Step by Step
Using Eq. (9) of Theorem 3, The volume of the parallelepiped spanned by ${\bf{u}} = \left( {1,0,0} \right)$, ${\bf{v}} = \left( {0,2,0} \right)$, and ${\bf{w}} = \left( {1,1,2} \right)$ is
$V = \left| {{\bf{u}}\cdot\left( {{\bf{v}} \times {\bf{w}}} \right)} \right| = \left| {\det \left( {\begin{array}{*{20}{c}}
{\bf{u}}\\
{\bf{v}}\\
{\bf{w}}
\end{array}} \right)} \right|$
$V = \left| {\det \left( {\begin{array}{*{20}{c}}
1&0&0\\
0&2&0\\
1&1&2
\end{array}} \right)} \right|$
$V = \left| {1\left( {2\cdot2 - 1\cdot0} \right) - 0\left( {0\cdot2 - 1\cdot0} \right) + 0\left( {0\cdot1 - 1\cdot2} \right)} \right|$
$V = \left| 4 \right| = 4$
