We introduce two useful objects, the Kronecker delta
↓
and the Levi-Civita tensor
↓
With these objects and the Einstein summation convention
↓ (in its naive form: any repeated index is summed over, rather than the one used in general relativity) we find
We can now write the scalar triple product
↓ very elegantly
and read of its symmetry under cyclic permutations of the three vectors.
The triple vector product
↓ also falls quite easily
If the vectors commute, we get the standard result