Let the vectors a,b,c represent three coterminous edges of a parallelopiped of volume V.

Question

Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $\vec{a},\vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to :

Accepted Answer

Given that the volume $V$ of the parallelepiped formed by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is represented by the scalar triple product $[\vec{a},\vec{b},\vec{c}]$, which is the determinant of the 3 x 3 matrix with vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as its rows (or columns).

When the vectors representing the coterminous edges of the parallelepiped are $\vec{a}$, $\vec{b}+\vec{c}$, and $\vec{a}+2\vec{b}+3\vec{c}$, the volume $V$ of the parallelepiped is represented by :

$\begin{gathered} V=[\vec{a},\vec{b}+\vec{c},\vec{a}+2\vec{b}+3\vec{c}] \ =|\begin{array}{ccc}1 \quad  0 \quad  0 \ 0 \quad  1 \quad  1 \ 1 \quad  2 \quad  3\end{array}|[\begin{gathered} \vec{a} \quad \vec{b} \quad \vec{c}] \ =1(3-2)[\vec{a},\vec{b},\vec{c}] \ =[\vec{a} \quad \vec{b} \quad \vec{c} \end{gathered}]=V \end{gathered}$

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