hi Kalid hope you have time:
so if i see c as a vector that can be split into components (a,b) i can just as well see the area defined by c as a vector that can be split into components of the areas defined by a and b.
so when we split the right triangle into two self-similar triangles, we really split the area into the components of its growth in the two orthogonal dimensions...
what do you think of this? i found this really beautiful but i just wonder how to think of it.
then what we see as the area is really the magnitude of the area vector just as the hypotenuse is just the magnitude of the vector c. and what is magnitude? in fact it is just projection in a different coordinate system..
somehow this is really whirling up quite a bit of my understanding of direction. pretty interesting..