Question

If $\overline{a}=2\overline{m}+\overline{n},\overline{b}=\overline{m}-2\overline{n}$, Angle between the unit vectors $\overline{m}$ and $\overline{n}$ is ${60}^o$. a, b are the sides of a parallelogram, then the lengths of the diagonals are?

• A. $\sqrt{7},\sqrt{5}$
• B. $\sqrt{13},\sqrt{5}$
• C. $\sqrt{7},\sqrt{13}$
• D. $\sqrt{11},\sqrt{13}$

Answer 1

The correct option is C $\sqrt{7},\sqrt{13}$

Diagonals of parallelogram are

$\begin{array}{c}\overrightarrow{a}+\overrightarrow{b}=3\hat{m}-\hat{n}\\ \overrightarrow{a}-\overrightarrow{b}=\hat{m}-3\hat{n}\end{array}$

$\begin{array}{c}{\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2=\left(3\hat{m}-\hat{n}\right).\left(3\hat{m}-\hat{n}\right)\\ =10-\hat{n}.\hat{n}=10-6\times \frac{1}{2}=7\end{array}$

${\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2=\sqrt{7}$

${\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2=\left(3\hat{m}+\hat{n}\right).\left(3\hat{m}\hat{n}\right)$

$=10-\hat{m}.\hat{n}=10+6\times \frac{1}{2}=13$

${\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2=\sqrt{13}$

Then,

We get $C$ is correct answer.