Consider a quadrilateral whose sides are given by the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$, such that $\vec{a}+\vec{b}+\vec{c}+\vec{d}=0$.
I'm told that the area of the quadrilateral can be calculated by half of the determinant of the matrix with columns given by the diagonals of the quadrilateral.
Why is this the case?
My initial approach was to use the cross product to find the area of the parallelogram spanned by two sides of the quadrilateral but this didn't seem yield anything meaningful.
Best Answer
Consider an arbitrary quadrilateral.
Move two vertices parallelly to a diagonal, so that two sides become aligned with the other diagonal. This transformation does not change the area.
Then move a vertex so that one side becomes aligned with the first diagonal. This transformation also preserves the area.
The area is that of a triangle, half the cross-product of the diagonal vectors.