A vector is a point with a property of an added direction. In short: an arrow.
Nice drawings can be made using arrows over the whole plane. But doing this by hand is tedious. So we do it by code.
In Processing the thing to look at is a PVector.
There is a nice tutorial about PVectors:
http://processing.org/learning/pvector/
Vector on a plane (or in space) - when the vectors are ordered, that is vary in a "physical" way - form a so called vector field. The vectors are like hair being combed.
One way to imagine a vector field is to think of magnets, the field lines, so beautifully made visible by iron dust:
http://www.openprocessing.org/visuals/?visualID=21418
Another way is to look at fluids, waves:
http://www.openprocessing.org/visuals/?visualID=12569
This is also a nice example:
http://www.openprocessing.org/visuals/?visualID=6707
Or look at this one:
http://www.openprocessing.org/visuals/?visualID=27653
With color:
http://www.openprocessing.org/visuals/?visualID=1036
search on keyword "field" in OpenProcessing.org
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