Past exam of the mathematics course of the University of Cambridge / 2005 / ia / Paper 1 / 1C / v / Solution

Interchange the two dummy indices $j,k$. The product $a_j{a}_k$ is symmetric under this interchange, while the Levi-Civita symbol changes sign. Thus the contraction equals its negative and is the zero vector, equivalently $\mathbf{a}\times \mathbf{a}=0$.