Question

If the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are mutually perpendicular, then $\overrightarrow{a}\times \{\overrightarrow{a}\times$ {$\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b})\left\}\right\}=$

• A. ${\left|\overrightarrow{a}\right|}^2\overrightarrow{b}$
• B. $|\overrightarrow{b}{|}^4\overrightarrow{a}$
• C. ${\left|\overrightarrow{a}\right|}^4\overrightarrow{b}$
• D. $\left|\overrightarrow{a}\right|\overrightarrow{b}$

Answer 1

The correct option is C ${\left|\overrightarrow{a}\right|}^4\overrightarrow{b}$

$=\overline{a}\times (\overline{a}\times ((\overline{a}\cdot \overline{b})\overline{a}-|a{|}^2\overline{b}\left)\right)$
$=(\overline{a}\times \overline{b}\times \overline{b}(-|\overline{a}{|}^2\left)\right)$
$=\left(\right(\overline{a}-\overline{b})\overline{a}-|a{|}^2\overline{b})(-|a{|}^2)$
$=|a{|}^4\overline{b}$