Linear Algebra
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A vector of magnitude 1.
There does not seem to be an under vector room, but there is vector space. Vector space is a structure that is formed by a collection of vectors. This is a term in mathematics.
Vector systems are a branch of mathematics that is used to manipulate measurements that have a value as well as a direction. Common examples are velocity, acceleration, force, etc - measurements involving motion. However, some motion-related measurements are not vectors. Distance, speed are not.
In math, a "vector field" is an abstract term for a set, and a number of operations, that have specific properties. Matrices of the same size, for example, all 3 x 2 matrices, combined with matrix addition and multiplication by a scalar, happens to have all those properties. You may want to read an introductory Linear Algebra book for more details.