Euclidean Geometry #3

$\vec{AB}$ = (1,0,0) - (2,2,2) = (-1,-2,-2);

$\vec{AC}$ = (0,1,0) - (2,2,2) = (-2,-1,-2);

$\vec{AD}$ = (1,0,1) - (2,2,2) = (-1,-2,-1);

Volume of the parallepiped = | $\vec{AB}\cdot(\vec{AC}\times\vec{AD})$ | = absolute value of the dot product of $\vec{AB}$ and the cross product $\vec{AC}\times\vec{AD}$ = absolute value of the mixed product of $\vec{AB}$ , $\vec{AC}$ and $\vec{AD}$ =

| $\begin{aligned} \begin{vmatrix} -1 & -2 & -2 \\ -2 & -1 & -2 \\ -1 & -2 & -1 \end{vmatrix}\end{aligned}$ | = | - 1 - 4 - 8 + 2 + 4 + 4 | = |-3| = 3