Basis Vector and Unit Vector
A unit vectoris a vector whose norm is unity: ||||=1. That's all. Any non-zero vectorcan define a unit vector /||||.How to compute the unit vector
Compute the magnitude of vector a
https://math.jianshu.com/math?formula=%5C%7C%20%7B%5Cvec%20a%7D%5C%7C%20%3D%20%5Csqrt%20%7Bx%5E2%2By%5E2%2Bz%5E2%7DCompute the unit vector
https://math.jianshu.com/math?formula=vector%20%2F%20%5C%7C%20%7B%5Cvec%20a%7D%5C%7C 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.
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