Basis Vector and Unit Vector

RecursiveFrog

A unit vector  is a vector whose norm is unity: ||||=1. That's all. Any non-zero vector  can define a unit vector /||||.

How to compute the unit vector

Compute the magnitude of vector a
Compute the unit vector
i.e. divide the vector a into the magnitude of the vector a

A basis vector is one vector of a basis, and a basis has a clear definition: it is a family of linearly independent vectors which spans a given vector space. So both have nothing to do. Your confusion may come from the fact that basis vectors are usually chosen as unit vectors, for the sake of simplicity.

For example, (0,3) and (2,0) form a basis of the plane (seen as a -vector space). So both (0,3) and (2,0) are basis vectors. (1,0) is a unit vector, but not a basis vector in that case. But you could also consider another basis made of (0,1) and (1,0), then (1,0) would also be a unit vector.

A unit vector does not "do" anything (if we set dual spaces aside...). But there are operators, such as the inner product, which "do" some things.