## Parallelepiped

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 $$\alpha,\beta,\gamma \in\langle 0{,}1\rangle$$.)
The volume of a parallelepiped is the absolute value of the determinant whose columns are vectors $$\mathbf a,\mathbf b,\mathbf c$$, i.e. $$V=\Biggl|\det \begin{pmatrix} 3 & 2 & 2 \\ 1 & 1 & 3 \\ 1 & 1 & 2 \\ \end{pmatrix}\Biggr|=|-1|=1$$.