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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.

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closed as off-topic by David Masip, Toros91, Stephen Rauch, Icyblade, Aditya Jun 2 '18 at 1:49

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  • $\begingroup$ This question should be migrated to Mathematics. $\endgroup$ – JahKnows May 30 '18 at 8:54
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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.

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A Euclidean vector is any vector with a magnitude and direction.

A unit vector, is a vector with a magnitude 1.

So it is fair to say that all unit vectors are Euclidean vectors. However, not all Euclidean vectors are unit vectors.

Given the statement above we can conclude that any Euclidean vector in $\mathbb{R}^3$ can be described by the unit vectors $e_1, e_2, e_3$. These span the space.

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