 Exterior cube

The exterior cube of a vector space $V$ over a field is its third exterior power ${\Lambda }^{3}V$. It is generated by products $u\wedge v\wedge w$, multilinear in the three vectors and zero whenever two inputs coincide. Every alternating trilinear map out of $V^3$ factors uniquely through a linear map from ${\Lambda }^{3}V$.

If $e_1,\dots ,e_n$ is a basis, the products $e_i\wedge e_j\wedge e_k$ for $i<j<k$ form a basis, so $\dim {\Lambda }^{3}V=\left(\genfrac{}{}{0ex}{}{n}{3}\right)$. In particular it vanishes for $n<3$ and is one-dimensional for $n=3$.

In an oriented three-dimensional coordinate space,Thus the exterior product packages oriented volume; swapping two vectors reverses its sign. The alternating condition, including vanishing repeated inputs, remains the definition even in characteristic two, where minus and plus signs agree.