Question

If $\overrightarrow{m},\overrightarrow{n}$ are non-parallel unit vectors and $\overrightarrow{r}$ is a vector which is perpendicular to $\overrightarrow{m}$ and $\overrightarrow{n}$ such that $\left|\overrightarrow{r}\right|=5$ and ${\left|\overrightarrow{m}+\overrightarrow{n}\right|}^2=2+4\left|\overrightarrow{m}\times \overrightarrow{n}\right|$ , then the value of ${\left|\left[\overrightarrow{m}\overrightarrow{n}\overrightarrow{r}\right]\right|}^2$ is equal to

• A. $7$
• B. $\frac{21}{5}$
• C. $5$
• D. $\frac{10}{3}$

Answer 1

Answer: (C)

$\overrightarrow{r}$ is parallel to $\overrightarrow{m}\times \overrightarrow{n}$
Let $\theta$ be angle between $\overrightarrow{m}$ and $\overrightarrow{n}$
$\therefore {\left|\overrightarrow{m}+\overrightarrow{n}\right|}^2=2+4\left|\overrightarrow{m}\times \overrightarrow{n}\right|$
$1+1+2cos\theta =2+4\times sin\theta$
$\Rightarrow tan\theta =\frac{1}{2}$
$|[\overrightarrow{m}\overrightarrow{n}\overrightarrow{r}]{|}^2=|\overrightarrow{r}\cdot (\overrightarrow{m}\times \overrightarrow{n}){|}^2$
$=|\overrightarrow{r}{|}^2|\overrightarrow{m}{|}^2|\overrightarrow{n}{|}^2{\sin }^2\theta$
$=25\times \frac{1}{5}=5$