Euclidean Foundations: Direction Vectors, Translations, Parallelness, Betweenness, and Orientation

Direction (vectors), parallelness, translation

We'll now formalize the notion of a direction. We do this with vectors $\langle{a,b}\rangle$ ($a,b$ real numbers), which you can intuitively think of as arrows from $(0,0)$ to $(a,b)$. If $a,b$ are not both zero, we call the corresponding vector nonzero.

Then a direction is a set of nonzero vectors $\{\langle{a,b}\rangle \ne \langle{0,0}\rangle\}$ such that for any $\langle{a,b}\rangle$ and $\langle{c,d}\rangle$ in the set, there exists a positive real $t$ such that $ta = c$ and $tb = d$. (Intuitively, think dilation/scaling by a factor of $t$ about the origin $(0,0)$, to get from $(a,b)$ to $(ta,tb)$.) For example, "north" is $\{\langle{\mathbb{R}^+,0}\rangle\}$, "south" is $\{\langle{\mathbb{R}^-,0}\rangle\}$, and "northwest" is $\{\langle{-u,u}\rangle: u > 0\}$.

Often when being lazy, we'll specify a direction (vector) just by a single member of the set. For example, we'd represent the north, south, and northwest directions by $\langle{1,0}\rangle$, $\langle{-1,0}\rangle$, and $\langle{-1,1}\rangle$.

Given a head point $P_1 = (x_1,y_1)$ and a tail point $P_2 = (x_2,y_2)$, the vector $\overrightarrow{P_1P_2}$ from $P_1$ to $P_2$ is simply $\langle{x_2-x_1,y_2-y_1}\rangle$, which can intuitively be written as $P_2 - P_1$.

Two vectors are parallel if they have either the same or "opposite" (i.e. negative) directions. (In some contexts, they're called antiparallel if they correspond to opposite directions.)

A translation by the (translation) vector $v = \langle{a,b}\rangle$ is a "transformation" (mapping) sending the point $P = (x,y)$ to the point $P+v = (x+a,y+b)$. It's easy to check that if $\overrightarrow{AB}$ is parallel to $\overrightarrow{CD}$, then $\overrightarrow{(A+v)(B+v)}$ is parallel to $\overrightarrow{(C+v)(D+v)}$.

Betweenness and orientation (counterclockwise and clockwise)

In this section, we'll denote vectors/directions $\langle{a,b}\rangle$ by the shorthand $(a,b)$, when the context is clear.

Let's say $(x,y)$ is to the left of (or counterclockwise of) a nonzero $(a,b)$ iff $ay-bx \ge 0$, and strictly (to the) left (or strictly counterclockwise) iff $ay-bx > 0$. (What happens when $ay-bx = 0$? To gain intuition for this definition, try working it out for some examples, thinking in terms of "slopes" $b/a$, $y/x$, etc., at least when $a,x$ are nonzero.)

Similarly, say $(x,y)$ is to the right (or clockwise of) iff $ay-bx \le 0$, and strictly (to the) right (or (strictly) clockwise) iff $ay-bx < 0$. Note that if $(a,b)$ is left or right of $(c,d)$, then $(c,d)$ is right or left of $(a,b)$, respectively.

(Often left/right/etc. will actually mean strictly left/strictly right/etc., depending on context. But it's almost never a big deal, so don't worry too much about it.)

By orientation, we mean the relative leftness/rightness/etc. of vectors. For example, vectors $u,v,w$ are in counterclockwise orientation if $v$ is left of $u$ and $w$ is left of $v$, and in clockwise orientation if $v$ is right of $u$ and $w$ is right of $v$.

Let's try to define betweenness. We want a definition such that $(x,y)$ lies in between $(1,0)$ and $(0,1)$ iff $x,y > 0$; in between $(1,0)$ and $(1,1)$ iff $x-y,y > 0$, etc.

The following works: let $(a,b)$ and $(c,d)$ be two nonzero vectors, with $(c,d)$ left of $(a,b)$ (or equivalently, $(a,b)$ is right of $(c,d)$). Then $(x,y)$ is between $(a,b)$ and $(c,d)$ iff $(x,y)$ is left of $(a,b)$ and $(c,d)$ is left of $(x,y)$.

(Again, I recommend that you work through some examples to gain intuition.)