Cross product - example
Recall that in 3-dimensional space the cross product of two non-collinear vectors [math]\displaystyle{ {\bf a} }[/math] and [math]\displaystyle{ b }[/math] is defined as a vector [math]\displaystyle{ {\bf c} = {\bf a} \times {\bf b} }[/math] such that [math]\displaystyle{ {\bf c} }[/math] is perpendicular to the plane spanned by [math]\displaystyle{ {\bf a} }[/math] and [math]\displaystyle{ {\bf b} }[/math], the length of [math]\displaystyle{ {\bf c} }[/math] is equal to the are of parallelogram with sides [math]\displaystyle{ {\bf a} }[/math] and [math]\displaystyle{ {\bf b} }[/math], and the triple [math]\displaystyle{ {\bf a}, {\bf b}, {\bf c} }[/math] is positively oriented.
Consider a tetrahedron in [math]\displaystyle{ \mathbb R^3 }[/math]. For each face of this tetrahedron, draw a vector beginning at some point of the face, pointing outwards, and with length equal to the area of the face. What is the sum of these four vectors?