Are euclidian vectors and unit vectors same thing? Consider this statement : Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R3. Consider the vectors e1 = (1,0,0), e2 = (0,1,0) and e3 = (0,0,1). Then any vector in R3 is a linear combination of e1, e2 and e3.
A unit vector in a normed vector space is a vector of length 1. A unit vector is an euclidian vector of length 1. Not every euclidean vetor has length 1.