Question

A parallelogram is constructed on the vector $\overrightarrow{a}=3\overrightarrow{p}-\overrightarrow{q}$ and $\overrightarrow{b}=\overrightarrow{p}+3\overrightarrow{q}$, given that $\left|\overrightarrow{p}\right|=\left|\overrightarrow{q}\right|=2$ and the angle between $\overrightarrow{p}$ and $\overrightarrow{q}$ is $\frac{\pi }{3}$. The length of a diagonal is

• A. $4\sqrt{5}$
• B. $4\sqrt{3}$
• C. $4\sqrt{7}$
• D. none of these

Answer 1

The correct option is B $4\sqrt{3}$

The diagonals of the parallelogram are represented by the vectors

$a+b=(3p-q)+(p+3q)=4p+2q$

and $a-b=(3p-q)-(p+3q)=2p-4q$

Now, ${|a+b|}^2={|4q+2q|}^2=16{\left|p\right|}^2+4{\left|q\right|}^2+16p.q$

$=16{\left(2\right)}^2+4{\left(2\right)}^2+16\left(2\right)\left(2\right)\cos \frac{\pi }{3}$

$=64+16+32=112[\because \cos \frac{\pi }{3}=\frac{1}{2}]\Rightarrow |a+b|=\sqrt{112}=4\sqrt{7}$

Similarly, $|a-b|=4\sqrt{3}$

Hence the lengths of the diagonals are $4\sqrt{3}$ and $4\sqrt{7}$.