Question

The volume of the parallelopiped constructed on diagonals of the faces of the given rectangular parallelopiped is $m$ times the volume of the given parallelopiped. Then $m$ is equal to

Answer 1

Let the sides of given parallelopiped are $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},$ then its volume $\left(V_i\right)$ $=\left|\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]\right|$
sides of parallelopiped whose sides are the face diagonal of given parallelopiped are:
$(\overrightarrow{a}+\overrightarrow{b}),(\overrightarrow{b}+\overrightarrow{c}),(\overrightarrow{c}+\overrightarrow{a})$
Hence, its volume $\left(V_f\right)$ $=\left|\left[\begin{array}{ccc}\overrightarrow{a}+\overrightarrow{b}& \overrightarrow{b}+\overrightarrow{c}& \overrightarrow{c}+\overrightarrow{a}\end{array}\right]\right|$
$=2\left|\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]\right|=2V_i$
Therefore, $m=2$