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When reading about vectors I sometimes have seen unit vectors multiplied by the components and other times I've seen basis vectors used instead. $$v=x \hat i+y \hat j+z \hat k$$

$$v=xe_x+ye_y+ze_z$$ Occasionally, I've seen both used in a single source. As far as I can tell, they seen to be doing the same thing, i.e., showing what direction each component is pointing while not changing the numerical value of any of the components (This, at least to me, seems to be what a unit vector does).

My questions are: What is the difference between an unit vector and a basis vector? And are they interchangeable in specifying the directions of components?

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  • $\begingroup$ It's worth pointing out that it doesn't really make sense to talk about 'a basis vector': any nonzero vector is part of some basis (and in fact part of an infinite number of them), so it's not really helpful. Rather you need to talk about a particular basis, which is a set of vectors with nice properties (LI, span the space) of which the vector of interest is one, and this is what people mean usually. It is useful to talk about a unit vector since that is defined as a property of the vector on its own. $\endgroup$ – tfb Jun 23 '16 at 7:27
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A unit vector $v$ is a vector whose norm is unity: $||v||=1$. That's all. Any non-zero vector $w$ can define a unit vector $w/||w||$.

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 $\mathbb{R}$-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 last thing: 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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  • $\begingroup$ I don't know if mathematicians make a clear distinction between vectors and its components but shouldn't it be said that the vector having components (0,3) and (2,0) in the system of a particular basis could also be chosen to be basis vectors? $\endgroup$ – Dvij Mankad Jun 23 '16 at 4:51
  • $\begingroup$ Thank you for all your help. It makes much more sense now. $\endgroup$ – K Ferreira Jun 23 '16 at 13:16
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    $\begingroup$ @Dvij, (0,3) and (2,0) can not be the components of a vector. The components of a vector are scalars. If we say $V=(a_0, a_1, \ldots, a_n)$, then the $a_i$ are the components, and there is an implied basis; $B_0, B_1, \ldots, B_n$ such that $V=a_0B_0+a_1B_1,+\dots+a_nB_n$ $\endgroup$ – Solomon Slow Jun 24 '16 at 17:36
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A unit vector might be a basis vector, and vice-versa: a unit vector is simply a vector whose magnitude is 1, while a basis vector is an element of a basis of a vector space $V$, that is, a set of vectors that span (i.e. generate by means of their linear combinations) the vector space $V$.

in your first example the basis on which to expand the vectors of your space has been chosen to be the Cartesian basis $(\hat i, \hat j, \hat k)$ in which each basis vector is oriented in one of the three orthogonal directions $x,y,z$.

These basis vectors, in this case are also unit vectors, because it’s usually easier to work with unit vectors as basis vectors, but as it has already been pointed out in previous answers, it’s not at all a requirement for a basis vector, to be also a unit vector.

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