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 $\overrightarrow a.\overrightarrow b=\overrightarrow b.\overrightarrow c=\overrightarrow c.\overrightarrow a=0$