Question

If $\left|\overrightarrow{a}\right|=2$ and $\left|\overrightarrow{b}\right|=3$ and $\overrightarrow{a}.\overrightarrow{b}=0,$ then $(\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b}))))$

• A. $48\overrightarrow{b}$
• B. $-24\overrightarrow{b}$
• C. $48\overrightarrow{a}$
• D. $-48\overrightarrow{a}$

Answer 1

The correct option is A $48\overrightarrow{b}$

$\overrightarrow{a}.\overrightarrow{b}=0$
$\therefore \overrightarrow{a}$ and $\overrightarrow{b}$ are mutually perpendicular.
Also $\left|\overrightarrow{a}\right|=2$ and $\left|\overrightarrow{b}\right|=3$
Let $\overrightarrow{a}=2\hat{i}$ then $\overrightarrow{b}=3\hat{j}$
$\therefore (\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b}))))$
$=(2\hat{i}\times (2\hat{i}\times (2\hat{i}\times (2\hat{i}\times 3\hat{j}))))$
$=(2\hat{i}\times (2\hat{i}\times (2\hat{i}\times 6\hat{k})))$
$=(2\hat{i}\times (2\hat{i}\times 12\hat{j}))$
$=(2\hat{i}\times 24\hat{k})$
$=48\hat{j}=48\overrightarrow{b}$