Q.Show that vector area of a quadrilateral ABCD is $\frac{1}{2}(\overline{AC}\times\overline{BD})$, where AC and BD are its diagonals.

Split quadrilateral $ABCD$ into $\triangle ABC$ and $\triangle ACD$ using diagonal $AC$. With position

vectors $\bar a,\bar b,\bar c,\bar d$:

(as vector areas, i.e. before taking magnitudes). The total vector area is

$\frac12\overrightarrow{AB}\times\overrightarrow{AC}+\frac12\overrightarrow{AC}\times\overrightarrow{AD} =\frac12\overrightarrow{AC}\times(\overrightarrow{AD}-\overrightarrow{AB})$ …

Split the quadrilateral into two triangles by one diagonal AC, write each triangle's vector area as a cross product from the shared vertex A, add them, …

Sign errors when factoring AC out of the two cross-product terms (the order of the cross product must be …