## Verifying a Vector Identity (BAC-CAB) using Levi-Civita

Author: **CHRISTINE ADELLE L. RICO**

Here is another method of verifying a vector formula using the Levi-Civita symbol. Levi-Civita symbol is a tensor of rank three and is defined by +1 if the indices are in even permutation of , -1 if the indices are in odd permutation, and 0 if any two indices are the same.

**Prove that .**

*Proof:*

Let .

Using the Levi-Civita symbol, we rewrite the cross products,

.

Since each term is only scalar, they can be rearranged such that,

where is summed over.

Note that is only nonzero if are all different from . There are two possibilities of its implications, either,

and or

and .

Consider the case of and , which gives for any value of . If and , . Therefore,

.

We can now write the proof so that the th component is,

.

Therefore,

.

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Adelle is currently pursuing her MS Physics degree at the Mindanao State University- Iligan Institute of Technology in Iligan City.

June 16th, 2016 at 6:48 pm

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