Answer
$({\bf u}\times{\bf v}) \cdot{\bf w}= ({\bf v}\times{\bf w}) \cdot {\bf u} = 8$
(the equality is true)
$V=8$
Work Step by Step
${\bf u}\times{\bf v}={\bf 2i}\times{\bf 2j}=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
2 & 0 & 0\\
0 & 2 & 0
\end{array}\right|$
$=(0-0){\bf i}-(0-0){\bf j}+(4-0){\bf k}$
$=4{\bf k}$
$=\langle 0, 0, 4 \rangle$
${\bf w}={\bf k} = \langle 0, 0, 2 \rangle$
$({\bf u}\times{\bf v}) \cdot {\bf w}=0(0)+0(0)+4(2)= 8$
On the other hand,
${\bf v}\times{\bf w}={\bf 2j}\times{\bf 2k}=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
0 & 2 & 0\\
0 & 0 & 2
\end{array}\right|$
$=(4-0){\bf i}-(0-0){\bf j}+(0-0){\bf k}$
$=4{\bf i}$
$=\langle 4, 0, 0 \rangle$
${\bf u}= \langle 2, 0, 0 \rangle$
$({\bf v}\times{\bf w}) \cdot {\bf u}=4(2)+0(0)+0(0)=8.$
Thus,
$({\bf u}\times{\bf v}) \cdot{\bf w}= ({\bf v}\times{\bf w}) \cdot {\bf u} = 8.$
and the volume of the parallelepiped determined by the three vectors is
$V=|({\bf u}\times{\bf v}) \cdot {\bf w}|=8$
