Question

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

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

Answer 1

Answer: (C)

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