Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Question

Let $\vec{e_i}$ denote a unit vector. Then we can write:

$\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$,

where $\epsilon_{ijk}$ is the Levi Civita symbol.

Can someone intuitively explain me why this is true and what the formula tells me?

Answer 1

Consider an orthonormal basis $e_1,e_2,e_3$ for $\mathbb{R}^3$,we can write $a\times b$ as a uniqUe linear combination of $e_i\times e_j$(i,j=1,2,3) also we can write $e_i\times e_j$ as a unique combination in the basis $e_1,e_2,e_3$ with the form ${c}^1_{i,j}e_1+{c}^2_{i,j}e_2+{c}^3_{i,j}e_3$. Now we show ${c}^k_{i,j}$ with ${\epsilon}_{i,j,k}$ and called levi-civita symbol, ${\epsilon}_{i,j,k}$ is projection with sign of the vector $e_i\times e_j$ on $e_k$.

${\epsilon}_{i,j,k}=0$ if there are coinciding values of the indices i, j, k.

${\epsilon}_{i,j,k}=1$ if the values of the indices i, j, k form an even permutation of the numbers 1, 2, 3.

${\epsilon}_{i,j,k}=-1$ if the values of the indices i, j, k form an odd permutation of the numbers 1, 2, 3.