Question

A parallelogram is constructed on the vector $a=3 p-q$ and $b=p+3 q$, given that $|p|=|q|=2$ and the angle between $p$ and $q$ 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

Answer: (B)

The diagonals of the parallelogram are represented by the vectors.
$a+b=(3 p-q)+(p+3 q)=4 p+2 q$
and $a-b=(3 p-q)-(p+3 q)=2 p-4 q$
Now, $|a+b|^{2}=|4 q+2 q|^{2}$
$=16|p|^{2}+4|q|^{2}+16 p \cdot q$
$=16(2)^{2}+4(2)^{2}+16(2)(2) \cos \frac{\pi}{3}$
$=64+16+12=112$
$\left(\because \cos \frac{\pi}{3}=\frac{1}{2}\right)$
$\Rightarrow |a+b|=\sqrt{112}=4 \sqrt{7}$
Similarly, $|a-b|=4 \sqrt{3}$
Hence the lengths of the diagonals are $4 \sqrt{3}$ and $4 \sqrt{3}$.