Let a, b, c be coterminous edges of a parallelepiped of volume V.

Question

Let $\vec a,\vec b,\vec c$ be coterminous edges of a parallelepiped of volume $V$. The volume of the parallelepiped with coterminous edges $\vec a$, $\vec b+\vec c$, $\vec a+2\vec b+3\vec c$ is:

Accepted Answer

Using the determinant of coefficients relative to $\{\vec a,\vec b,\vec c\}$:
$$\begin{vmatrix}1&0&0\0&1&1\1&2&3\end{vmatrix} = 1(3-2) = 1$$

So the new volume $=1\times[\vec a\ \vec b\ \vec c] = V$.

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