Verifying a Vector Identity (BAC-CAB) using Levi-Civita | Quantum Science Philippines

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

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 $n$th 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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