Question

If a parallelogram is constructed on the vectors $\overrightarrow{a}=3\overrightarrow{p}-\overrightarrow{q},\overrightarrow{b}=\overrightarrow{p}+3\overrightarrow{q}$ and $|\overrightarrow{p}|=|\overrightarrow{q}|=2$ and angle between $\overrightarrow{p}$ and $\overrightarrow{q}$ is $\frac{\pi }{3}$ , then the ratio of the lengths of the sides is

• A. $\sqrt{7}:\sqrt{13}$
• B. $\sqrt{6}:\sqrt{2}$
• C. $\sqrt{3}:\sqrt{5}$
• D. $1:2$

Answer 1

The correct option is A $\sqrt{7}:\sqrt{13}$

We have

$\overrightarrow{a}=3\overline{p}-\overline{q}$

$\overrightarrow{b}=\overline{p}+3\overline{q}$

So

$|\overline{a}|=\sqrt{9|\overline{p}{|}^2+|\overline{q}{|}^2-6\overline{p}\cdot \overline{q}}$

$=\sqrt{40-12}$

$=2\sqrt{7}$

$|\overline{b}|=\sqrt{|\overline{p}{|}^2+9|q{|}^2+6\overline{p}\cdot \overline{q}}$

$=2\sqrt{13}$

$\therefore \frac{|\overline{a}|}{|\overline{b}|}=\frac{\sqrt{7}}{\sqrt{13}}$