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
${\bf u}\times{\bf v}=({\bf i+j-2k})\times({\bf -i-k})=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
1 & 1 & -2\\
-1 & 0 & -1
\end{array}\right|$
$=(-1-0){\bf i}-(-1-2){\bf j}+(0+1){\bf k}= -{\bf i} +3{\bf j} +{\bf k}$
$=\langle-1, 3, 1 \rangle$
${\bf w}={\bf 2i+4j-2k} = \langle 2, 4, -2 \rangle$
$({\bf u}\times{\bf v}) \cdot {\bf w}=-1(2)+3(4)+1(-2)= 8$
${\bf v}\times{\bf w}=({\bf -i-k})\times({\bf 2i+4j-2k})=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
-1 & 0 & -1\\
2 & 4 & -2
\end{array}\right|$
$=(0+4){\bf i}-(2+2){\bf j}+(-4-0){\bf k}= 4{\bf i} -4{\bf j}-4{\bf k}$
$=\langle 4, -4, -4 \rangle$
${\bf u}= \langle 1, 1, -2 \rangle$
$({\bf v}\times{\bf w}) \cdot {\bf u}=4(1)-4(1)-4(-2)= 8$
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
$V=|({\bf u}\times{\bf v}) \cdot {\bf w}|=8$
