Question

If the volume of the parallelepiped with $\left(a\times b\right),\left(b\times c\right),\left(c\times a\right)$ as coterminous edges is $8$ cubic units, then the volume of the parallelepiped with $\left(\left(a\times b\right)\times \left(b\times c\right)\right),\left(\left(b\times c\right)\times \left(c\times a\right)\right),$ and $\left(\left(c\times a\right)\times \left(a\times b\right)\right)$ as coterminous edges is?

• A. $8$ cubic units
• B. $512$ cubic units
• C. $64$ cubic units
• D. $24$ cubic units

Answer 1

The correct option is C

$64$ cubic units

Explanation for the correct option :

Find the volume of the parallelepiped,

A parallelepiped has three edges as $\left(a\times b\right),\left(b\times c\right),\left(c\times a\right)$, and its volume is $8$ cubic units. So $\left[\left(a\times b\right)\left(b\times c\right)\left(c\times a\right)\right]=8$.

Now the volume of the parallelepiped with edges $\left(\left(a\times b\right)\times \left(b\times c\right)\right),\left(\left(b\times c\right)\times \left(c\times a\right)\right),$ and $\left(\left(c\times a\right)\times \left(a\times b\right)\right)$ is given as: $\left[\left(\left(a\times b\right)\times \left(b\times c\right)\right)\left(\left(b\times c\right)\times \left(c\times a\right)\right)\left(\left(c\times a\right)\times \left(a\times b\right)\right)\right]$.

It is known that $\left[a\times bb\times cc\times a\right]={\left[abc\right]}^2$.

So the volume of the required parallelepiped is given as:

$\begin{array}{rcl}\left[\left(\left(a\times b\right)\times \left(b\times c\right)\right)\left(\left(b\times c\right)\times \left(c\times a\right)\right)\left(\left(c\times a\right)\times \left(a\times b\right)\right)\right]& =& {\left[\left(a\times b\right)\left(b\times c\right)\left(c\times a\right)\right]}^2\\ & =& 8^2\\ & =& 64\end{array}$

Hence, the correct option is C.