Let a parallelopiped be formed on the three vectors$\overset{\rightarrow}{OA}=\overset{\rightarrow}{a}, \overset{\rightarrow}{OB}=\overset{\rightarrow}{b}$and $\overset{\rightarrow}{OC}=\overset{\rightarrow}{c}$Now, $\overset{\rightarrow}{b}\times\overset{\rightarrow}{c}=bc\sin 90^{\circ}\hat{n}=bc\hat{n}$where $\hat{n}$ is unit vector along $\overset{\rightarrow}{OA}$ perpendicular to the plane containing $\overset{\rightarrow}{b}$ and $\overset{\rightarrow}{c}.$Now $\overset{\rightarrow}{a}.\left(\overset{\rightarrow}{b}\times\overset{\rightarrow}{c}\right)$ $=\overset{\rightarrow}{a}. bc\hat{n}=(a)(bc)\cos 0^{\circ}=abc$ which is equal in magnitude to the volume of the parallelopiped.