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# A parallelogram has adjacent sides $\overrightarrow a=3\overrightarrow p-\overrightarrow q\:\;and\:\:\overrightarrow b=\overrightarrow p+3\overrightarrow q$ where $|\overrightarrow p|=|\overrightarrow q|=2$, and the angle between $\overrightarrow p\:\:and\:\:\overrightarrow q$ is $\large\frac{\pi}{3}$. The length of its diagonals are?

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If adjacent sides of a parallelogram are $\overrightarrow a\:\:and\:\:\overrightarrow b$ then
its diagonals are $\overrightarrow a+\overrightarrow b\:\:and\:\:\overrightarrow a-\overrightarrow b$
$\therefore\:\overrightarrow a+\overrightarrow b=(3\overrightarrow p-\overrightarrow q)+(\overrightarrow p+3\overrightarrow q)$
$=4\overrightarrow p+2\overrightarrow q$
and
$\overrightarrow a-\overrightarrow b=(3\overrightarrow p-\overrightarrow q)-(\overrightarrow p+3\overrightarrow q)$
$=2\overrightarrow p-4\overrightarrow q$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b|^2=16|\overrightarrow p|^2+4|\overrightarrow q|^2+16|\overrightarrow p||\overrightarrow q|cos\large\frac{\pi}{3}$
$=64+16+32=112$
$\therefore \:|\overrightarrow a+\overrightarrow b|=\sqrt {112}=4\sqrt 7$
and
$\Rightarrow\:|\overrightarrow a-\overrightarrow b|^2=4|\overrightarrow p|^2+16|\overrightarrow q|^2-16|\overrightarrow p||\overrightarrow q|cos\large\frac{\pi}{3}$
$=16+64-32=48$
$\therefore \:|\overrightarrow a-\overrightarrow b|=\sqrt {48}=4\sqrt 3$
$\therefore\:$ the length of diagonals are $4\sqrt 3\:and \:4\sqrt 7$
answered Dec 13, 2013