'Vector graphics is the use of geometrical primitives such as points, lines, curves, and shapes or polygon(s), which are all based on mathematical expressions, to represent images in computer graphics.
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Vector graphics are based on vectors (also called paths, or strokes) which lead through locations called control points. Each of these points has a definite position on the x and y axes of the work plan. Each point, as well, is a variety of database, including the location of the point in the work space and the direction of the vector (which is what defines the direction of the track). Each track can be assigned a color, a shape, a thickness and also a fill. This does not affect the size of the files in a substantial way because all information resides in the structure; it describes how to draw the vector.'
Vectors are made up of shapes as opposed to pixels, so you can blow them up as large as you want, with no resultant loss in quality. Wikipedia's information on vector art is a bit technical. A working knowledge of how to produce vectors is useful to fully understand them, just as a working knowledge of how to properly expose a photograph helps piece together the technical jargon of photography. I don't wish to present too much of the technical aspect here, but instead display my favorite vectors from around deviantART for your viewing pleasure.
There are many reasons to love vectors. Enjoy adorable shirts? A large number of deviants have Threadless accounts for their designs, and many of the pieces below can be purchased there. A couple of these deviants take design requests as well.