Area of a Paralellogram
The area of a paralellogram is one of the most elementary formulars in computational geometry.
Here are some examples of its application:
 Calculating the area of a triangle or an arbitrary (convex and concave) polygon
 Test if a point lies left, right or on a line (test if three points are ordered clockwise (CW) or counter clockwise (CCW))
 Test if a polygon is convex or concave
 Test if a point lies inside a cone given by three points
 Test if two lines intersect without calculating the intersection point
 Sorting points by "leftness" as a preparing step for Graham's convex hull algorithm
The area of a paralellogram spanned by two vectors a and b can be expressed in R^{2} by the 2 x 2 determinant:
A(P) =

a_{x} a_{y}
b_{x} b_{y}

= a_{x} * b_{y}  a_{y} * b_{x}

It turns out that this can easily be shown geometrically:
(For simplicity we will assume that all coordinates are positive. I leave it as an exercise for the reader to show that the equation holds for arbitrary cooridnates.)
We will prove the area calculation by a disjoint decomposition of the bounding box around a paralellogram:
As it can be easily seen the area of the parallelogram is what remains if we substract from the red boudingbox the four triangles (yellow, green, orange, light turquoise) and the two rectangles in the upperleft and lowerright corner.
Viewed mathematically we have got:
 area of red bouding box = (a_{x} + b_{x}) * (a_{y} + b_{y})
 green and light tourquise triangle added together form the rectangle: a_{x} * a_{y}
 yellow and orange triangle added together form the rectangle: b_{x} * b_{y}
 upperleft and lower rigth rectangle are each a rectangle of b_{x} * a_{y}, thus togher: 2 * a_{y} * b_{x}
Now we put all pieces into a formula and reduce:
A(P) = (a_{x} + b_{x}) * (a_{y} + b_{y})  (a_{x} * a_{y} + b_{x} * b_{y} + 2 * a_{y} * b_{x} )
= a_{x} * a_{y} + a_{x} * b_{y} + a_{y} * b_{x} + b_{x} * b_{y}  a_{x} * a_{y}  b_{x} * b_{y}  2 * a_{y} * b_{x}
= a_{x} * a_{y} + a_{x} * b_{y} + a_{y} * b_{x} + b_{x} * b_{y}  a_{x} * a_{y}  b_{x} * b_{y}  2 * a_{y} * b_{x}
= a_{x} * b_{y}  a_{y} * b_{x}