# How do you establish the concept of direction with vectors?

My introduction to vectors was as arrows pointing in a particular direction and they were described by relatively simple formalisms based on Euclidean geometry. At this point, I'm aware that this isn't an adequate description, particularly in dimensions above 3. My understanding now is that vectors are lists, in a nutshell. My question, with that in mind is, if the arrow formalism falls short, how do you describe vector "directions"? If this seems vague or unintelligible, I guess I'm looking for something as intuitive as the vector formalism but more accurate.

Vector direction is an equivalence relation on the set of nonzero vectors, which is defined as follows: two vectors $\vec v$, $\vec w$ have the same direction if and only if there exists a positive real number $r>0$ such that $r \vec v = \vec w$.

So, for example, $\vec v = \langle -1,3,7,2 \rangle$ and $\vec w = \langle -3,9,21,6 \rangle$ have the same direction because $3 \vec v = \vec w$.

To prove that this relation is an equivalence relation:

• $1 \vec v = \vec v$;
• if $r \vec v = \vec w$ then $\frac{1}{r} \vec w = \vec v$;
• if $r \vec v = \vec w$ and $s \vec w = \vec u$ then $(rs) \vec v = \vec u$.

From the way this equivalence relation is defined, one can see that the equivalence class of a vector $\vec v$, meaning the set of all vectors $\vec w$ having the same direction as $\vec v$, is just the set of all positive scalar multiples of $\vec v$. Let me define this equivalence class to be the direction of $\vec v$.

The direction of $\vec v$ can be understood from several geometric perspectives.

First, if you place $\vec v$ so that it is based at the origin, and if you draw the ray $R$ based at the origin and passing through the tip of $\vec v$, then a vector $\vec w$ has the same direction as $\vec v$ if and only if, when $\vec w$ is places so that it is based at the origin, the ray $R$ also passes through the tip of $\vec w$.

Second, draw the ray $R$ as above, and let $p$ be the point where $R$ pierces the unit sphere $S^{n-1}$. This gives a one-to-one correspondence between the set of all directions and the set of points on $S^{n-1}$.

So, for example, in the plane this gives a one-to-one correspendence between the set of all directions and the set of points on the unit circle $S^1$. Sometimes people like to parameterize the points of $S^1$ by their angle, assigning angle values in $[0,2\pi)$, and so this gives a one-to-one correspondence between the set of directions and the set $[0,2\pi)$. But, that's not really quite as good as the one-to-one correspondence with $S^1$ itself, because there is a break of continuity as you approach $2\pi$ from below.

Vectors are more abstract than just lists. You'll see that when you study more mathematics. Thinking of a vector as an arrow with length and direction doesn't always make sense.

For vectors in Euclidean space it does. Then the direction of a vector is essentially the directed line through the origin that contains it, so that the vectors $v \ne 0$ and $\lambda v$ have the same direction when $\lambda > 0$. Sometimes you want to think of the direction of $v$ as the unit vector $(1/|v|) v$ where $|v|$ is the length of $v$.

• That's fair. Let's say we were looking at a non-euclidean geometry, though. Like spherical, perhaps. What would a vector direction be in that sense? Or hyperbolic geometry? Or even without a geometric representation, just, like as an à la carte topic in algebra? Commented Oct 17, 2017 at 0:19
• Those noneuclidean geometries are "curved spaces" so they are not vector spaces. But near each point they are "almost flat" so you have directions at each point, but the directions vary from point to point in a subtle way. This material belongs to the field of differential geometry. Commented Oct 17, 2017 at 0:27

"In every dimension vectors can be considered as arrows (i) pointing in a particular direction (ii)."

This is just an intuitive notion based on the fact that a vector always determines a straight line (i) (you being in $\mathbb{R}^2$ or in $\mathbb{R}^{4563}$), and (ii) the direction of a vector determines which orientation of the straight line its demtermines you're looking at.

But this notions could be substituted by thinking of vectors as any point at some vector space. Note that this vector space could be anyone, so the notion of vectors as a "list", in my opinion, is pretty much biased... as it leads you to think just in cartesian vector spaces, that is, vectors of the form $v=(a_1,a_2,...)$. But the (many) others like "the vector space of functions", "the vector space of matrix", etc. are not achieved by this intuitive notion.