Happy New Year!
I came across this conundrum when reviewing 3D vectors.
I'm really baffled by the solution:
The only part I don't understand in the solution is how the equation
Area ABFE = Area ABCD + Area CDEF + Area ADE - Area BCF
gets derived.
There doesn't seem to be a straightforward geometric illustration. Could anyone offer some help?
There is a straight-forward geometric reasoning here. If you have two regions $X, Y$ in the plane then we always have $$ \operatorname{Area}X=\operatorname{Area}Y +\operatorname{Area}(X\setminus Y) -\operatorname{Area}(Y\setminus X) $$ In this case, they have described $\operatorname{Area}Y$ as the sum of the two parallelograms, while each of the two set differences are the triangles you see.