Question

Vector $\overrightarrow{a}$ & $\overrightarrow{b}$ make an angle $\theta =\frac{2\pi }{3}$. If $|\overrightarrow{a}|=1,|\overrightarrow{b}|=2$ then ${\{(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})\}}^2=$

• A. $225$
• B. $250$
• C. $275$
• D. $300$

Answer 1

The correct option is D $300$

${\left\{\left(\overrightarrow{a}+\overrightarrow{3b}\right)\times \left(\overrightarrow{3a}-\overrightarrow{b}\right)\right\}}^2$

$={\left\{3\overrightarrow{a}\times \overrightarrow{a}-\overrightarrow{a}\times \overrightarrow{b}+9\overrightarrow{b}\times \overrightarrow{a}-3\overrightarrow{b}\times \overrightarrow{b}\right\}}^2$

$={\left\{3\times 0-\overrightarrow{a}\times \overrightarrow{b}-9\overrightarrow{a}\times \overrightarrow{b}-3\times 0\right\}}^2$

$={\left\{-10\overrightarrow{a}\times \overrightarrow{b}\right\}}^2$

$=100{\left\{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \right\}}^2$

$=100{\left[2\sin \frac{2\pi }{3}\right]}^2$

$=100{\left[2\sin \left(\pi -\frac{\pi }{3}\right)\right]}^2$

$=100{\left[2\sin \frac{\pi }{3}\right]}^2$

$=100{\left(2\times \frac{\sqrt{3}}{2}\right)}^2$

$=100\times 3$

$=300$