## Thomas' Calculus 13th Edition

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