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 what is the volume of the parallelepiped with \[\left( {\left( {\left( {a \times b} \right) \times \left( {b \times c} \right)} \right),\left( {\left( {b \times c} \right) \times \left( {c \times a} \right)} \right),\left( {\left( {c \times a} \right) \times \left( {a \times b} \right)} \right)} \right)\] as coterminous edges?
A. 8 cubic units
B. \[512\] cubic units
C. \[64\] cubic units
D. \[24\] cubic units

Answer 1

Hint: Use the formula of the volume of the parallelepiped and find the volume of both parallelepipeds.

Formula Used:
The volume of a parallelepiped formed by three vectors \[a,b,c\] is: \[\left[ {a b c} \right] = \left| {a \cdot \left( {b \times c} \right)} \right|\]

Complete step by step solution:
Given:
The volume of the parallelepiped with coterminous edges \[\left( {a \times b} \right),\left( {b \times c} \right),\left( {c \times a} \right)\] is 8 cubic units.
Coterminous edges of another parallelepiped are \[\left( {\left( {\left( {a \times b} \right) \times \left( {b \times c} \right)} \right),\left( {\left( {b \times c} \right) \times \left( {c \times a} \right)} \right),\left( {\left( {c \times a} \right) \times \left( {a \times b} \right)} \right)} \right)\]
 Apply the formula of the volume of the parallelepiped.
Then the volume of the first parallelepiped is,
\[\left[ {\left( {a \times b} \right) \left( {b \times c} \right) \left( {c \times a} \right)} \right] = 8\] cubic units \[.....\left( 1 \right)\]

The volume of the second parallelepiped is,
\[V = \left[ {\left( {\left( {a \times b} \right) \times \left( {b \times c} \right)} \right) \left( {\left( {b \times c} \right) \times \left( {c \times a} \right)} \right) \left( {\left( {c \times a} \right) \times \left( {a \times b} \right)} \right)} \right]\]
\[ \Rightarrow \]\[V = {\left[ {\left( {a \times b} \right) \left( {b \times c} \right) \left( {c \times a} \right)} \right]^2}\]
Substitute the value of equation \[\left( 1 \right)\] in the above equation.
\[V = {\left[ 8 \right]^2}\]
\[ \Rightarrow \]\[V = 64\] cubic units

Hence the correct option is C.

Note:Students often get confused with the concept of a parallelepiped. A parallelepiped is a polyhedron whose six faces are parallelograms.