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

If $\overrightarrow a=2\hat i+\hat j-2\hat k\:\;and\:\:\overrightarrow b=\hat i+\hat j$ and $\overrightarrow c$ is such that $\overrightarrow c.\overrightarrow a=|\overrightarrow c|$, $|\overrightarrow c-\overrightarrow a|=2\sqrt 2$ and the angle between $\overrightarrow a\times\overrightarrow b$ and $\overrightarrow c$ is $30^{\circ}$, then $|(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c|=?$

$\large\frac{2}{3} \\ \large\frac{3}{2} \\ 2 \\ 3 $

1 Answer

Given: $\overrightarrow a=2\hat i+ \hat j-2\hat k,\:\overrightarrow b=\hat i+\hat j,\:\overrightarrow a.\overrightarrow c=|\overrightarrow c|$ and $|\overrightarrow c-\overrightarrow a|=2\sqrt 2$
$\Rightarrow\:|\overrightarrow c-\overrightarrow a|^2=|\overrightarrow c|^2+|\overrightarrow a|^2-2\overrightarrow c.\overrightarrow a=8$
$\Rightarrow\:|\overrightarrow c|^2+9-2|\overrightarrow c|=8$
$\Rightarrow\:|\overrightarrow c|^2-2\overrightarrow c|+1=0$
$\Rightarrow\:|\overrightarrow c|=1$
$|(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c|=|\overrightarrow (\overrightarrow a\times\overrightarrow b)|.|\overrightarrow c|.sin\theta$
where $\theta$ is the angle between $\overrightarrow a\times\overrightarrow b\:\:and\:\:\overrightarrow c$ which is given to be $30^{\circ}$
$\overrightarrow a\times\overrightarrow b=2\hat i-2\hat j+\hat k$
$\therefore\:|(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c|=3 sin 30^{\circ}=\large\frac{3}{2}$
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