What is Vector Art?

How do we say that an artwork is a vector? What is the difference between a vector and vexel? A couple of minutes ago I browsed my acquaintance's facebook account where he posted his "vector art". He traced the photos of his friends using photoshop and it looked like comic versions of the photos. In the image he traced a thick black lines for the outlines and blurred or smudged out the skin. I kinda looked like this self-portait that I made years ago.

He also used brushes to create smokey eyes and shadows for the lighting. I just want to clarify to all digital art the misuse of the term vector. Even I myself have misconceptions on vectors. Here is another so-called vector art. Most of the elements are created using photoshop brushes.

Quoted from a forum in dA,
Vector is made using mathematical equations instead of pixels so no matter how far you zoom in it recalculates and and looks perfect instead of a pixel based system. That's the technical definition however it has a specific style attached to it. It is usually either super simplistic or hyper realistic. As for vexel it is technically pixel based art that looks like vector but this too is know for hyper realism.
Therefore, what we see above are not vector art. If it is created using brushes and several tools such as blur or smudge, it is not vector. Vectors are created using the pen tool and shape tools alone. It means when we interpolate (increase the resolution of) a vector image, the line will still look smooth. Usually the file format of vectors are .ai and .eps. On the other hand, we have vexels. Vexels are rasterized versions of vectors. They are usually the jpegs or gifs that looks like vectors but the quality will be lost when we interpolate it. This is one of my favorite and the only vexel art that i made using adobe illustrator and then edited it in photoshop.

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