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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Vector Algebra
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Q)

If $\overrightarrow a\:and\:\overrightarrow b$ are two non collinear unit vectors and $\overrightarrow u=\overrightarrow a-(\overrightarrow a.\overrightarrow b)\overrightarrow b\:\:and\:\:\overrightarrow v=\overrightarrow a\times\overrightarrow b$, then $|\overrightarrow v|=?$

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A)
Given: $\overrightarrow v=\overrightarrow a\times\overrightarrow b$
$\Rightarrow\:|\overrightarrow v|^2=|\overrightarrow a\times\overrightarrow b|^2=|\overrightarrow a|^2|\overrightarrow b|^2 sin^2 \theta.$
$\Rightarrow\:|\overrightarrow v|^2=sin^2\theta$................(i) (Since $\overrightarrow a\:\:and\:\:\overrightarrow b$ are unit vectors.)
Also given : $\overrightarrow u=\overrightarrow a-(\overrightarrow a.\overrightarrow b)\overrightarrow b$
$|\overrightarrow u|^2=|\overrightarrow a|^2+(\overrightarrow a.\overrightarrow b)^2.|\overrightarrow b|^2-2(\overrightarrow a.\overrightarrow b)^2$
$=1+cos^2\theta-2cos^2\theta=1-cos^2\theta=sin^2\theta$
$i.e.,|\overrightarrow u|^2=sin^2\theta$...........(ii)
From $(i)\:\;and\:\:(ii)$ $ |\overrightarrow v|=|\overrightarrow u|$
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