Question

If $\overrightarrow{x}=\overrightarrow{a}+\overrightarrow{b},\overrightarrow{y}=\overrightarrow{a}-\overrightarrow{b},|\overrightarrow{a}|=2,|\overrightarrow{b}|=3$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$ ,then $|\overrightarrow{x}\times \overrightarrow{y}|$ is equal to

• A. $5\sqrt{3}$
• B. $6$
• C. $4\sqrt{3}$
• D. $9$
• E. $6\sqrt{3}$

Answer 1

Answer: (E)

$|\overrightarrow{x}\times \overrightarrow{y}|=|(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b})|$
$=|-\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}|$
$=2|\overrightarrow{a}\times \overrightarrow{b}|$
$=2|\overrightarrow{a}||\overrightarrow{b}|\sin \frac{\pi }{3}$
$=2\times 2\times 3\times \frac{\sqrt{3}}{2}$
$=6\sqrt{3}$