# Find the volume of the parallelepiped with one vertex at the origin and adjacent

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at $\left(1,3,0\right),\left(-2,0,2\right),\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left(-1,3,-1\right)$.

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mhalmantus

Setup the matrix A, where the columns are the vectors given.
$A=\left[\begin{array}{ccc}1& -2& -1\\ 3& 0& 3\\ 0& 2& -1\end{array}\right]$
The volume of the parallelepiped is equal to the absolute value of the determinant of A.
Volume $=|det\left(A\right)|$
So lets calculate the determinant. Expanding along the first column (because ${A}_{3,1}=0$ gives us two terms, but i'll show it for clarification)gives us to shown calculation.
$A=|\begin{array}{ccc}1& -2& -1\\ 3& 0& 3\\ 0& 2& -1\end{array}|=|\begin{array}{cc}0& 3\\ 2& -1\end{array}|-3|\begin{array}{cc}-2& -1\\ 2& -1\end{array}|+0|\begin{array}{cc}-2& -1\\ 0& 3\end{array}|=-6-12=-18$
Volume of the parallelepiped is 18.
Results:
Volume of the parallelepiped is 18.