Question

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