Question

If the volume of the parallelopiped formed by three non-coplanar vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is 4 cubic units, then $[\overrightarrow{a}\times \overrightarrow{b}$ $\overrightarrow{b}\times \overrightarrow{c}$ $\overrightarrow{c}\times \overrightarrow{a}]=$

• A. 64
• B. 16
• C. 4
• D. 8

Answer 1

The correct option is B 16

Since the volume of a parallelepiped formed by three non-coplanar vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is $4$ cubic units, $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=4$

$[\overrightarrow{a}\times \overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}\overrightarrow{c}\times \overrightarrow{a}]$

$=(\overrightarrow{a}\times \overrightarrow{b}).\left[\right(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{c}\times \overrightarrow{a}\left)\right]$

$=(\overrightarrow{a}\times \overrightarrow{b}).\left[\right((\overrightarrow{b}\times \overrightarrow{c}).\overrightarrow{a})\overrightarrow{c}-((\overrightarrow{b}\times \overrightarrow{c}).\overrightarrow{c}\left)\overrightarrow{a}\right]$

$=(\overrightarrow{a}\times \overrightarrow{b}).\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{c}$

$=\overrightarrow{c}.(\overrightarrow{a}\times \overrightarrow{b})\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

$=[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}{]}^2$

$=16$