Question

If $\overline{a}$,$\overline{b}$,$\overline{c}$ are unit vectors such that $\overline{a}\cdot (\overline{b}\times \overline{c})=4$, then $[\overline{a}\times \overline{b},\overline{b}\times \overline{c},c\times \overline{a}]$ is

• A. $16$
• B. $64$
• C. $4$
• D. $8$

Answer 1

The correct option is A $16$

Consider $[\overline{a}\times \overline{b}\overline{b}\times \overline{c}\overline{c}\times \overline{a}]$
$=(\overline{a}\times \overline{b})\cdot [\overline{k}\times (\overline{c}\times \overline{a}\left)\right]$ where $k=\overline{b}\times \overline{c}$ $=\overline{a}\times \overline{b}\cdot \left[\right(\overline{k}\cdot \overline{a})\overline{c}-(\overline{k}\cdot \overline{c}\left)\overline{a}\right]$
$=(\overline{a}\times \overline{b})\cdot \left[\right[(\overline{b}\times \overline{c})\cdot \overline{a}]\overline{c}-[(\overline{b}\times \overline{c})\cdot \overline{c}\left]\overline{a}\right]$
$=(\overline{a}\times \overline{b})\cdot \left[\right(\overline{b}\times \overline{c}\cdot \overline{a}]\overline{c}-0$
$=\left(\right(\overline{b}\times \overline{c})\cdot \overline{a})\left[\right(\overline{a}\times \overline{b})\cdot \overline{c}]$
$=[\overline{a}\cdot (\overline{b}\times \overline{c}\left)\right]\left[\right(\overline{a}\times \overline{b})\cdot \overline{c}][\overline{c}\cdot (\overline{a}\times \overline{b}]$
$=[\overline{a}\cdot (\overline{b}\times \overline{c}){]}^2=16$