Editing Cross product (section)

=== Using Levi-Civita tensors ===
* In any basis, the cross-product <math>a \times b</math> is given by the tensorial formula <math>E_{ijk}a^ib^j</math> where <math>E_{ijk} </math> is the covariant [[Levi-Civita symbol#Levi-Civita tensors|Levi-Civita]] tensor (we note the position of the indices). That corresponds to the intrinsic formula given [[#As an external product|here]].
* In an orthonormal basis '''having the same orientation as the space''', <math>a \times b</math> is given by the pseudo-tensorial formula <math> \varepsilon_{ijk}a^ib^j</math> where <math>\varepsilon_{ijk}</math> is the Levi-Civita symbol (which is a pseudo-tensor). That is the formula used for everyday physics but it works only for this special choice of basis.
* In any orthonormal basis, <math>a \times b</math> is given by the pseudo-tensorial formula <math>(-1)^B\varepsilon_{ijk}a^ib^j</math> where <math>(-1)^B = \pm 1</math> indicates whether the basis has the same orientation as the space or not.

The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.