Answer
$({\bf u}\times{\bf v}) \cdot{\bf w}= ({\bf v}\times{\bf w}) \cdot {\bf u} = -7$
(the equality is true)
$V=7$
Work Step by Step
${\bf u}\times{\bf v}=({\bf 2i+j})\times({\bf 2i-j+k})=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
2 & 1 & 0\\
2 & -1 & 1
\end{array}\right|$
$=(1-0){\bf i}-(2-0){\bf j}+(-2-2){\bf k}= {\bf i} -2{\bf j} -4{\bf k}$
$=\langle 1, -2, -4 \rangle$
${\bf w}={\bf i+2k} = \langle 1, 0, 2 \rangle$
$({\bf u}\times{\bf v}) \cdot {\bf w}=1(1)-2(0)-4(2)= -7$
${\bf v}\times{\bf w}=({\bf 2i-j+k})\times({\bf i+2k})=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
2 & -1 & 1\\
1 & 0 & 2
\end{array}\right|$
$=(-2-0){\bf i}-(4-1){\bf j}+(0+1){\bf k}= -2{\bf i} -3{\bf j}+{\bf k}$
$=\langle-2, -3, 1 \rangle$
${\bf u}= \langle 2, 1, 0 \rangle$
$({\bf v}\times{\bf w}) \cdot {\bf u}=-2(2)-3(1)+1(0)= -7$
Thus,
$({\bf u}\times{\bf v}) \cdot{\bf w}= ({\bf v}\times{\bf w}) \cdot {\bf u} = -7.$
so the volume of the parallelepiped determined by the three vectors is
$V=|({\bf u}\times{\bf v}) \cdot {\bf w}|=7$
