Question:
If $\vec{A}=2 \vec{\imath}+3 \vec{\jmath}+4 \vec{k}$ and $\vec{B}=4 \vec{\imath}+3 \vec{\jmath}+2 \vec{k}$, find $\vec{A} \times \vec{B}$
Solution:
A, $B$ and $C$ are mutually perpendicular vectors. Now, if we take cross product between any two vectors, the resultant vector will be in parallel to the third vector, as there are only three axis perpendicular to each other.
So if we consider $(A \times B)$, then it is parallel to $C$, and so angle between the resultant vector and $C$ is $0^{\circ}$, and $\sin \left(0^{\circ}\right)=0$. So, $C \times(A \times B)=0$