I was reading a blog post on [a simple derivation of the cross product](http://behindtheguesses.blogspot.com/2009/04/dot-and-cross-products.html). I learned how to determine the area of a [parallelogram enclosed by two vectors $A$ and $B$](http://4.bp.blogspot.com/_D1sP-NndkqU/SfDq13JeaCI/AAAAAAAAHHc/Q_IyEv4AXkY/s1600-h/cross.gif).


First, here is the [proof](http://1.bp.blogspot.com/_D1sP-NndkqU/SfDq4A_2oSI/AAAAAAAAHHk/lWBYwv6Mh4s/s1600-h/decomp.gif) of the solution ($area = A\_x B\_y - B\_x A\_y$)


And here's the [geometric implication](http://4.bp.blogspot.com/_D1sP-NndkqU/SfDrt4Z-ZEI/AAAAAAAAHHs/yYqx4O3SCjQ/s1600-h/3dvec.gif) of the solution.


It bewilders me that the geometric solution is simple but nonintuitive.


One commenter (Tim Poston) on the blog offered an explanation by using the following lemma, which he states can be easily shown using elementary plane geometry, without the use of right angles or base\*height to find areas:


$C(u+w,v) = C(u,v) + C(w,v)$


where $u, v, w$ are arbitrary vectors, and $C$ is the area function of the parallelogram between 2 vectors.


Here's my illustration of an [example implication of this lemma](https://i.imgur.com/RleJK.png) (visually simplified to triangles without loss of generality)


I was unable to prove this lemma, so I couldn't follow the argument. Any thoughts or direction with this lemma or the original problem would be appreciated.



Why does the area function of a parallelogram have a nonintuitive geometric solution?
It is better to see this property if you stay with parallelograms and use the Cavalieri Principle. Here is [page](http://matematica-para-todos.wikispaces.com/Propriedade+de+Determinante+por+Cavalieri) with an animation showing what I mean (clink on the link and accept).


There is also a demonstration of the first formula using triangles and again using only Cavalieri. There was an animation "proof without words" for it in the site "art of problem solving" but I can't find it anymore. 


