Verifying a Vector Identity (BAC-CAB) using Levi-Civita | Quantum Science Philippines
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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 \epsilon_{ijk} is a tensor of rank three and is defined by +1 if the indices i,j,k are in even permutation of 1,2,3, -1 if the indices are in odd permutation, and 0 if any two indices are the same.

Prove that \vec{a} \times (\vec{b}\times\vec{c}) = \vec{b} (\vec{a}\cdot\vec{c}) - \vec{c} (\vec{a}\cdot\vec{b}).

Proof:

Let \vec{a} = a_i, \vec{b} = b_j, \vec{c} = c_k.

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

(\vec{b}\times\vec{c})_m = b_j c_k \epsilon_{jkm}

(\vec{a}\times(\vec{b}\times\vec{c})_m)_n = a_i \epsilon_{imn} b_j c_k \epsilon_{jkm}.

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

(\vec{a}\times(\vec{b}\times\vec{c})_m)_n = a_i b_j c_k \epsilon_{imn} \epsilon_{jkm} where m is summed over.

Note that \epsilon_{imn} \epsilon_{jkm} is only nonzero if i,n,j, k are all different from m. There are two possibilities of its implications, either,

i=j and n=k or

i=k and n=j.

Consider the case of i=j  and n=k, which gives \epsilon_{imn} \epsilon_{inm} = -1 for any value of m. If i=k and n=j, \epsilon_{imn} \epsilon_{nim} = +1. Therefore,

\epsilon_{imn} \epsilon_{jkm} = \delta_{ik} \delta_{nj} - \delta_{ij} \delta_{nk}.

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

\begin{array} {rcl} (\vec{a}\times(\vec{b}\times\vec{c}))_n &=& a_i b_j c_k (\delta_{ik} \delta_{nj} - \delta_{ij} \delta_{nk})\\&=& a_i b_j c_k \delta_{ik} \delta_{nj} - a_i b_j c_k \delta_{ij} \delta_{nk}\\&=& a_i b_n c_i - a_i b_i c_n\\&=& b_n a_i c_i - c_n a_i b_i\\ &=& \vec{b} (\vec{a}\cdot \vec{c}) - \vec{c} (\vec{a} \cdot \vec{b})\end{array}.

Therefore,

\vec{a} \times (\vec{b}\times\vec{c}) = \vec{b} (\vec{a}\cdot\vec{c}) - \vec{c} (\vec{a}\cdot\vec{b}).

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

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