Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three vectors such that $\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]=4$ then value of $\left[\begin{array}{ccc}\overrightarrow{a}\times \overrightarrow{b}& \overrightarrow{b}\times \overrightarrow{c}& \overrightarrow{c}\times \overrightarrow{a}\end{array}\right]$ is

• A. $8$
• B. $16$
• C. $64$
• D. None of these

Answer 1

Answer: (B)

$16$

$\left[\begin{array}{ccc}\overrightarrow{a}\times \overrightarrow{b}& \overrightarrow{b}\times \overrightarrow{c}& \overrightarrow{c}\times \overrightarrow{a}\end{array}\right]$
$=\overrightarrow{a}\times \overrightarrow{b}.[(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{c}\times \overrightarrow{a})]$
$=\overrightarrow{a}\times \overrightarrow{b}\cdot [(\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a})\overrightarrow{c}-(\overrightarrow{b}\times \overrightarrow{c}.\overrightarrow{c})\overrightarrow{a}]$
$=\overrightarrow{a}\times \overrightarrow{b}\cdot \overrightarrow{c}\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$
$={\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]}^2$