Classification and Division

Vectors

What do map coordinates, airplanes, stock values and rabbits have in common? They can all be represented by vectors. Despite being pure-mathematical abstract constructs, vectors constitute one of the most useful and influential human inventions ever. Geometric vectors are the ones most familiar to us: we recall countless arrows on a blackboard from high-school physics classes, and the important lesson that speed is different from velocity because the latter has two quantities: magnitude and direction. Geometric vectors can also be understood in terms of perpendicular components: points on a Cartesian plane or on a map are therefore all geometric vectors; these have affected all of modern physics and are used in nearly all technology today. A natural extension of the idea of representing something in terms of multiple one-dimensional components is the algebraic vector, which in its purest form is merely an ordered n-tuple or a collection of real numbers which can be represented horizontally in parentheses or vertically as a matrix. Experimenting with these new vectors led to the invention of linear transformations and matrix operations. Converting what was once two-dimensional geometry into higher-dimensional math allowed mathematicians to use other sophisticated techniques from calculus and algebra in understanding these vectors. Every piece of data in a computer is an algebraic vector, and in the Theory of General Relativity every point in the universe throughout the history of time is an algebraic vector (instead of two quantities, they have four: length, width, depth and time); the usefulness of these is undisputable. Even higher-minded mathematicians then developed abstract vectors: anything meeting an unbelievably broad, abstract set of conditions is considered part of a vector space. Depending on your conditions, anything from elephants to colours to 19th-century poets to your own thoughts can be considered abstract vectors. Ultimately the brilliance of the vector, whether geometric, algebraic or abstract, is that one object can represent multiple pieces of information; objects like these have become so much a part of our thoughts that we never pause to consider or appreciate them.