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# The vectors $\overrightarrow a,\overrightarrow b,\overrightarrow c\:\overrightarrow d$ be are such that $(\overrightarrow a\times\overrightarrow b)\times (\overrightarrow c\times\overrightarrow d)=0$ and $P_1,\:P_2$ are two planes containing the pair of vectors $\overrightarrow a,\overrightarrow b$ and $\overrightarrow c,\overrightarrow d$ respectively, then the angle between the planes is ?

$\large (a)\:\:0\:\:\:\qquad\:\:(b)\:\:\frac{\pi}{4}\:\:\:\qquad\:\:(c)\:\:\frac{\pi}{3}\:\:\:\qquad\:\:(d)\:\:\frac{\pi}{2}$
Can you answer this question?

Toolbox:
• Normal to a plane is $\perp$ all the vectors on the plane.
Since it is given that $\overrightarrow a,\overrightarrow b$ lie on the plane $P_1$,
Normal to $P_1$ is $\perp$ to both $\overrightarrow a\:\:and\:\:\overrightarrow b$.
$\Rightarrow\:\overrightarrow n_1=\overrightarrow a\times\overrightarrow b$
Similarly normal to $P_2$= $\overrightarrow n_2$ is $\perp$ to $\overrightarrow c\:\:and \:\:\overrightarrow d$
$\Rightarrow\:\overrightarrow n_2=\overrightarrow c\times\overrightarrow d$
Given that $(\overrightarrow a\times\overrightarrow b)\times (\overrightarrow c\times\overrightarrow d)=0$
$\Rightarrow\:\overrightarrow n_1\times\overrightarrow n_2=0$
$\Rightarrow$ Both the planes are parallel.
$\therefore$ The angle is $0$.
answered Jan 5, 2014