Unit Vector vs. Basis Vector

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

ANSWER: Consider a "basis" vector as a "foundation" of a PARTICULAR Vector Space V. In other words, if a ⊂︎ V, then you must be able to fully DESCRIBE it in V. Example: can you fully "describe" a 3D vector accurately in 2 space? No. You don't have a sufficient "foundation" or basis to do so. Your basis MUST have at LEAST 3 ORTHOGONAL components to make sense for a 3 dimensional vector. Traditionally, we use i, j, and k as our basis vectors in 3D Vector Spaces. Yeah, it's that simple BUT Mathematicians INSIST on clarity! Now UNIT vectors are often the same as the basis vectors but need not be, it's function is different. A Unit Vector is just that, a "unit" that ALL vectors in V can use so they are all speaking the same DISTANCES (metrics). The "easiest" way to generate units is by dividing all vectors in the space BY their lengths! Equivalently, we can multiply each one by their Cosine. So why the big deal? Mathematicians versus Physicists and Engineers is why! Mathematicians, who usually come up with this stuff DEMAND precision! Engineers and Physicists tend to say," C'mon already!" Skip the "whys" we TRUST you, that's what you DO. Let us take the ball and we'll run with it!"