and there is an object coinciding with this property: the Levi-Civita symbol, denoted by ε. The Levi-Civita symbol entries can be represented by the Cartesian basis: which geometrically corresponds to the volume of a cube spanned by the orthonormal basis vectors, with sign indicating orientation (and not a "positive or negative volume"). Here, the orientation is fixed by ε123 = +1, for a right-handed system. A left-handed system would fix ε123 = −1 or equivalently ε321 = +1. The scalar triple product can now be written: with the geometric interpretation of volume (of the parallelepiped spanned by a, b, c) and algebraically is a determinant: This in turn can be used to rewrite the cross product of two vectors as follows: Contrary to its appearance, the Levi-Civita symbol is not a tensor, but a pseudotensor, the components transform according to: Therefore, the transformation of the cross product of a and b is: and so a × b transforms as a pseudovector, because of the determinant factor. The tensor index notation applies to any object which has entities that form multidimensional arrays – not everything with indices is a tensor by default. Instead, tensors are defined by how their coordinates and basis elements change under a transformation from one coordinate system to another. Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector. Applications of the δ tensor and ε pseudotensor Other identities can be formed from the δ tensor and ε pseudotensor, a notable and very useful identity is one that converts two Levi-Civita symbols adjacently contracted over two indices into an antisymmetrized combination of Kronecker deltas: The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn are used extensively in physics and engineering. For instance, it is clear the dot and cross products are distributive over vector addition: without resort to any geometric constructions – the derivation in each case is a quick line of algebra. Although the procedure is less obvious, the vector triple product can also be derived. Rewriting in index notation: and because cyclic permutations of indices in the ε symbol does not change its value, cyclically permuting indices in εkℓm to obtain εℓmk allows us to use the above δ-ε identity to