Cross product

Unlike the dot product, which returns a number, the cross product returns a vector and exists only in \(\mathbb R^3\). Its length is \(|\mathbf u||\mathbf v|\sin\theta\), where \(\theta\) is the angle between the vectors; hence it is zero exactly when \(\mathbf u\) and \(\mathbf v\) are parallel and largest when they are perpendicular. The area of a triangle with sides \(\mathbf u\) and \(\mathbf v\) is half that length.

The determinant in the formula is symbolic: it is expanded along the first row, with the vectors \(\mathbf i,\mathbf j,\mathbf k\) as entries. For example, \(\langle1,2,3\rangle\times\langle4,5,6\rangle=\langle-3,6,-3\rangle\). The minus sign of the middle term is the most frequent mistake when computing it.

Order matters: \(\mathbf v\times\mathbf u=-(\mathbf u\times\mathbf v)\), so the product is anticommutative and, in particular, \(\mathbf u\times\mathbf u=\mathbf 0\). It distributes over addition and pulls out scalars, but it is not associative: \(\mathbf i\times(\mathbf i\times\mathbf j)=-\mathbf j\), whereas \((\mathbf i\times\mathbf i)\times\mathbf j=\mathbf 0\).

Its main use will be to give the normal of a plane from two directions lying in it, and later the normal vector of a parametric surface, which is the piece needed to integrate over surfaces.