Quoted from:

Practical Handbook of Digital Mapping Terms and Concepts.

S. L. Arlinghaus; R. F. Austin; W. C. Arlinghaus, W. D. Drake, J. D. Nystuen

pp. 253-254.

vector

a mathematical entity showing both distance and
direction. In a mapping context, this word has come to refer to a
data structure for processing and displaying graphic data, represented
as strings of coordinates given the true position of features and their
boundaries. In contrast to the raster format, in which reference
is to pixels or matrix entries, the vector format is capable of stringing
together pieces of line, end-to-end, in order to create a curve that appears
smoother than does its raster counterpart formed from little squares (current
shape of a pixel). Vector data define complex geometric entities
that can be manipulated on the basis of attribut data; they are very
useful in this regard.

From a theoretical standpoint, the distinction
between vector and raster is parallel to the approach in linear algebra
that distinguishes between combining (using rules of algebra) vectors directly,
or combining (using rules of algebra) matrices that represent these vectors.
Thus, one place to look for theoretical material related to the problem
of shifting from vector to raster format is in the literature of modern
linear algebra texts that emphasize a transformational approach (rather
than the first-half of the twentieth century approach of "matrix algebra").