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# For non zero vectors $\overrightarrow a,\overrightarrow b\:and\:\overrightarrow c,\:\:if\:\:|(\overrightarrow a\times\overrightarrow b).\overrightarrow c|=|\overrightarrow a||\overrightarrow b||\overrightarrow c|$ then ?

Given : $|(\overrightarrow a\times\overrightarrow b).\overrightarrow c|=|\overrightarrow a||\overrightarrow b||\overrightarrow c|$
$\Rightarrow\:|\overrightarrow a||\overrightarrow b||\overrightarrow c|.sin\theta_1.cos\theta_2=|\overrightarrow a||\overrightarrow b||\overrightarrow c|$
where $\theta_1$ is angle between $\overrightarrow a\:and\:\overrightarrow b$ and
$\theta_2$ is angle between $\overrightarrow a\times\overrightarrow b\:\:and\:\:\overrightarrow c$
$\Rightarrow\:sin\theta_1.cos\theta_2=1$
$\Rightarrow\:\theta_1=\large\frac{\pi}{2}$ and $\theta_2=0$
$i.e.,$ Angle between $\overrightarrow a\:\:and\:\:\overrightarrow b$ Is $\large\frac{\pi}{2}$ and that
between $(\overrightarrow a\times\overrightarrow b)\:\:and\:\:\overrightarrow c$ is $0$
$\Rightarrow\:\overrightarrow a.\overrightarrow b=0.....(i)$ and
$\overrightarrow a\times\overrightarrow b$ is along the direction of $\overrightarrow c$
$\Rightarrow\:\overrightarrow c$ is $\perp$ to $\overrightarrow a\:\:and\:\:\overrightarrow b$
$\Rightarrow\:\overrightarrow a.\overrightarrow c=\overrightarrow b.\overrightarrow c=0.........(ii)$
From $(i)$ and $(ii)$ we get $\overrightarrow a.\overrightarrow b=\overrightarrow b.\overrightarrow c=\overrightarrow c.\overrightarrow a=0$