Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $|\overrightarrow{a}|=3,|\overrightarrow{b}|=2$ and angle between $\overrightarrow{a}$ and $\overrightarrow{b}\text{is}\frac{\pi }{3},$ then the area of the triangle with adjacent sides $\overrightarrow{a}+2\overrightarrow{b}$ and $2\overrightarrow{a}+\overrightarrow{b}$ in sq. units is

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

Answer 1

The correct option is C $\frac{9\sqrt{3}}{2}$

Area of triangle with adjacent sides: $\overrightarrow{a}+2\overrightarrow{b}\text{and}2\overrightarrow{a}+\overrightarrow{b}\text{is}=\frac{1}{2}\cdot |(\overrightarrow{a}+2\overrightarrow{b})\times (2\overrightarrow{a}+\overrightarrow{b})|=\frac{1}{2}\cdot |(\overrightarrow{a}\times 2\overrightarrow{a})+(\overrightarrow{a}\times \overrightarrow{b})+(2\overrightarrow{b}\times 2\overrightarrow{a})+(2\overrightarrow{b}\times \overrightarrow{b})|=\frac{1}{2}\cdot |0+(\overrightarrow{a}\times \overrightarrow{b})+4(\overrightarrow{b}\times \overrightarrow{a})+0|=\frac{1}{2}\cdot |4(\overrightarrow{b}\times \overrightarrow{a})-(\overrightarrow{b}\times \overrightarrow{a})|=\frac{1}{2}\cdot |3(\overrightarrow{b}\times \overrightarrow{a})|=\left|\frac{-3}{2}\right(\overrightarrow{a}\times \overrightarrow{b}\left)\right|$
Angle between $\overrightarrow{a}\text{and}\overrightarrow{b}=\frac{\pi }{3}$
$=\left|\frac{-3}{2}\right||a||b|\left|\sin \frac{\pi }{3}\right|=\frac{3}{2}\times 3\times 2\times \frac{\sqrt{3}}{2}=\frac{9\sqrt{3}}{2}$