Vector graphics/images use geometrical objects such as points, lines, shapes, and/or polygons. They are based on mathematical formulae to represent images in computer graphics. store these objects (geometrical primitives) as a mathematical formulae. The formulae allow you to construct the image to the best quality possible, according to your screen resolution. It uses the formulae to decide how best to display/scale the image.
The computer screen is made up of grids of individual pixels. Bitmap images have a definite pixel size, so as you scale it, it will affect the overall quality of the image. If you enlarge it, these pixels become magnified and will appear 'grainy'. Vector graphics, on the other hand, decides where to place the dots for the best results, when displaying the image. You can scale them to any size and still maintain the details and a high quality.
The following is a comparison between two similar images, the left being vector and the right being bitmap:
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