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# If $\overrightarrow a=2\hat i+\hat j-2\hat k\:\:and\:\:\overrightarrow b=\hat i+\hat j.$ and $\overrightarrow c$ is a vector such that $\overrightarrow a.\overrightarrow c=|\overrightarrow c|,\:\:|\overrightarrow c-\overrightarrow a|=2\sqrt 2$ and angle between $\overrightarrow a\times\overrightarrow b\:\:and\:\:\overrightarrow c$ is $\large\frac{\pi}{6}$, then $|(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c|=?$

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A)
Given: $\overrightarrow a.\overrightarrow c=|\overrightarrow c|\:\:and\:\:|\overrightarrow c-\overrightarrow a|=2\sqrt 2$
$\Rightarrow\:|\overrightarrow c-\overrightarrow a|^2=8$
$\Rightarrow\:|\overrightarrow c|^2+|\overrightarrow a|^2-2\overrightarrow c.\overrightarrow a=8$
$\Rightarrow\:|\overrightarrow c|^2+9-2|\overrightarrow c|=8$
(Since it is given that $\overrightarrow a=2\hat i+\hat j-2\hat k,\:\:\:|\overrightarrow a|=3$)
$\therefore |\overrightarrow c|=1$
Now
$(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c=|\overrightarrow a\times\overrightarrow b||\overrightarrow c|sin\large\frac{\pi}{6}$
(Since it is given that the angle between $\overrightarrow a\times\overrightarrow b\:\:\:and\:\:\:\overrightarrow c$ is $\large\frac{\pi}{6}$.)
$\Rightarrow\:(\overrightarrow a\times\overrightarrow b)\times\overrightarrow c=\large\frac{1}{2}$$|\overrightarrow a\times\overrightarrow b| =\large\frac{1}{2}$$|2\hat i-2\hat j+\hat k|=\large\frac{3}{2}$
Since $\overrightarrow a\times\overrightarrow c=2\hat i-2\hat j+\hat k$