$\vec{b} \times \vec{c}=b c \sin 90^{\circ} \hat{n}=b c \hat{n}$ where $\hat{n}$ is the unit vector along $\overrightarrow{O A}$ perpendicular to the plane containing $\vec{b}$ and $\vec{c}$.
$\vec{a} \cdot(\vec{b} \times \vec{c})=\vec{a} \cdot b c \hat{n}=(a)(b c) \cos 0^{\circ}$
$(\because \hat{n} \| \vec{a})=a b c=$ volume of the parallelopiped.