Parallelepiped
Task number: 2607
Calculate the volume of the parallelepiped determinde by vectors \(\mathbf a^T=(3{,}1,1)\), \(\mathbf b^T=(2{,}1,1)\) and \(\mathbf c^T=(2{,}3,2)\).
(A parallelepiped in \(\mathbb R^3\) contains points that can be expressed as a linear combination \(\alpha \mathbf a + \beta \mathbf b + \gamma \mathbf c\), where \(\alpha,\beta,\gamma \in\langle 0{,}1\rangle\).)
Solution
The volume of a parallelepiped is the absolute value of the determinant whose columns are vectors \(\mathbf a,\mathbf b,\mathbf c\), i.e. \(V=\Biggl|\det \begin{pmatrix} 3 & 2 & 2 \\ 1 & 1 & 3 \\ 1 & 1 & 2 \\ \end{pmatrix}\Biggr|=|-1|=1\).
