Question

$\overrightarrow{a},\overrightarrow{b}$ are such that $|\overrightarrow{a}|=\sqrt{3},|\overrightarrow{b}|=2$ and $(\overrightarrow{a},\overrightarrow{b})=\frac{\pi }{3}$. Then the area of the triangle with adjacent sides $\overrightarrow{a}+2\overrightarrow{b}$ and $2\overrightarrow{a}+\overrightarrow{b}$ is

• A. $5\sqrt{3}$
• B. $15$
• C. $\frac{9}{2}$
• D. $\frac{15}{2}$

Answer 1

The correct option is C $\frac{9}{2}$

Area of $△=\frac{1}{2}|(\overrightarrow{a}+\alpha \overrightarrow{b})\times (2\overrightarrow{a}+\overrightarrow{b})|$
$=\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}+4\overrightarrow{b}\times \overrightarrow{a}|$
$=\frac{1}{2}3|\overrightarrow{a}\times \overrightarrow{b}|$
$=\frac{1}{2}\times 3\times \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta$
$=3\times \sqrt{3}\times \frac{\sqrt{3}}{2}=\frac{9}{2}$