Determinants and the vector properties of length, area, volume

Length, area, volume, and higher dimensions of these values have vector associated with them. There is a notion of "negative area", in that it is a "space in a different direction". This property is necessary to preserve 0, from linearity, so that you can add two equal but opposite components and obtain 0.

The directionality of area/volume can also be seen from 2D and 3D cross products, when considering the result to be the magnitude of the area of the parallelogram/parallelepiped that is formed.

Examples include current density, dl in Biot-Savart Law, dA in Gauss' Law.

http://www.askamathematician.com/2013/05/q-why-are-determinants-defined-the-weird-way-they-are/