Visual walkthrough — Points, lines, line segments, rays — notation and differences

We are going to grow every object in the parent topic out of one tiny dot, using pictures. By the end you will see why a ray needs $t\ge 0$, a line needs "all $t$", and a segment needs $0\le t\le 1$ — not as rules to memorise, but as things you can watch happen.

Nothing here assumes you already know vectors, parameters, or coordinates. We build each idea the moment we need it.

---

Step 1 — Start with a single point

WHAT: We place one point and give it a name, $P$.

WHY: Every richer object (ray, line, segment) is going to be made of points. So we must start from the smallest possible thing — the "atom" of geometry.

PICTURE: Below, $P$ sits alone on the board. Notice there is no direction and no distance yet — just a location.

Recall Why can't a point have size?

If it had size ::: we could cut it in half and find something smaller inside — but a point is defined to be the smallest markable thing, so it must have no size.

---

Step 2 — Add a second point to earn a direction

WHAT: Place a second point $A$, and draw the little journey "from $P$ toward $A$".

WHY: Direction is the ingredient that turns a lonely dot into something we can travel along. We cannot build a ray or line without knowing which way to go.

PICTURE: The pale-yellow arrow shows the direction from $P$ to $A$. This arrow is our new tool.

WHY subtract? Subtraction answers "what do I add to $P$ to reach $A$?" — the leftover after removing $P$'s position is exactly the displacement.

---

Step 3 — Slide along the direction with a dial called $t$

WHAT: We write the position of the sliding dot as a formula: $\underset{\text{where you are now}}{\underbrace{(x,y)}}=\underset{\text{where you started}}{\underbrace{P}}+\underset{\text{dial value}}{\underbrace{t}}\underset{\text{one step of direction}}{\underbrace{\vec{v}}}.$

WHY a dial? We want one formula that can produce every point of a ray, a line, or a segment. The only difference between those three objects will turn out to be which values of $t$ we allow. So we need a single tunable knob first.

Let us read what different dial values do:

• $t=0$: you are at $P+0\cdot \vec{v}=P$. You have not moved.
• $t=1$: you are at $P+1\cdot \vec{v}=A$. You have moved exactly to $A$.
• $t=2$: you are at $P+2\vec{v}$ — twice as far, past $A$, same direction.
• $t=\frac{1}{2}$: you are halfway between $P$ and $A$.

PICTURE: Watch the dot at several dial settings. Each labelled tick is a different value of $t$.

Recall What point does

$t=1$ always give? $P+1\cdot \vec{v}=P+(A-P)=A$ ::: it lands exactly on the second point $A$.

---

Step 4 — Allow only $t\ge 0$: this is the ray

WHAT: Keep the dial but forbid negative settings. We allow every $t$ from $0$ upward: $\overrightarrow{PA}=\{\underset{\text{endpoint}}{\underbrace{P}}+t\vec{v}\mid \underset{\text{forward only}}{\underbrace{t\ge 0}}\}.$

WHY $t\ge 0$? Negative $t$ would push the dot behind $P$ (opposite to $\vec{v}$). A ray must start at $P$ and never leave from the back — so we cut off everything below $0$. The value $t=0$ is allowed, which is why the starting point $P$ is included; that starting point is the ray's single endpoint.

Notation: $\overrightarrow{PA}$ — one arrow, pointing away from the endpoint $P$, through the direction-marker $A$.

PICTURE: The forward half is solid pale-yellow (the ray). The backward half is faded and crossed out — it is not part of $\overrightarrow{PA}$.

---

Step 5 — Allow every $t$: this is the line

WHAT: Allow $t$ to be any real number. "Any real number" is written $t\in \mathbb{R}$ — the symbol $\mathbb{R}$ just means "the whole number line: negatives, zero, positives, fractions, everything." $\overleftrightarrow{PA}=\{P+t\vec{v}\mid \underset{\text{both directions, no limit}}{\underbrace{t\in \mathbb{R}}}\}.$

WHY every $t$? A line is the pure idea of straightness with no boundary. Since negative $t$ walks backward and positive $t$ walks forward, letting $t$ be anything removes both ends at once — no start, no finish.

Notation: $\overleftrightarrow{PA}$ — a double-headed arrow, because it is infinite on both sides.

PICTURE: Both halves are now solid. Arrowheads on each end signal "keeps going".

---

Step 6 — Trap $t$ between $0$ and $1$: this is the segment

WHAT: Allow only $0\le t\le 1$: $\overline{PA}=\{P+t\vec{v}\mid \underset{\text{start at }P,\text{stop at }A}{\underbrace{0\le t\le 1}}\}.$

WHY these bounds? From Step 3 we know $t=0$ gives $P$ and $t=1$ gives $A$. Everything strictly between ($0<t<1$) is inside the segment. We forbid $t<0$ (no going behind $P$) and $t>1$ (no going past $A$). Both endpoints are included because $t=0$ and $t=1$ are allowed.

Notation: $\overline{PA}$ — a plain bar, no arrows, because nothing runs to infinity.

PICTURE: Only the stretch between the two dots is drawn, both ends capped.

---

Step 7 — Degenerate case: what if the two points coincide?

WHAT: If $A=P$, then $\vec{v}=A-P=(0,0)$ — the zero vector, an arrow with no length and no direction.

WHY it matters: Plug it in: $P+t\vec{v}=P+t(0,0)=P$ for every $t$. The dial does nothing; the dot never moves.

Consequences, case by case:

• Ray / line: undefined — with no direction there is nothing to point along. You cannot draw $\overrightarrow{PA}$ or $\overleftrightarrow{PA}$ from a point to itself.
• Segment: $\overline{PP}$ collapses to the single point $P$; its length is $\sqrt{0^2+0^2}=0$.

PICTURE: The lonely dot with a crossed-out zero-arrow — the whole construction shrinks back to Step 1.

---

The one-picture summary

Same formula $(x,y)=P+t\vec{v}$ every time — only the allowed range of the dial $t$ changes. That single idea produces all three objects.

Object	Notation	Dial range	Endpoints
Point	$P$	(no dial)	—
Ray	$\overrightarrow{PA}$	$t\ge 0$	one ($P$)
Line	$\overleftrightarrow{PA}$	$t\in \mathbb{R}$	none
Segment	$\overline{PA}$	$0\le t\le 1$	two ($P,A$)

Recall Feynman retelling — say it back in plain words

Start with one dot, $P$. It's just a spot, no size, no direction. Put a second dot $A$ nearby; now there's a direction — a little arrow $\vec{v}$ pointing from $P$ toward $A$. Imagine a knob called $t$: turning it slides a moving dot along that arrow. At $t=0$ you're on $P$; at $t=1$ you're on $A$; bigger $t$ goes past $A$; negative $t$ goes behind $P$. Now the whole trick: if I only let the knob go from $0$ upward, the dot shoots forward forever — that's a ray. If I let the knob be any number, it runs off both ways — that's a line. If I trap the knob between $0$ and $1$, it starts at $P$ and stops at $A$ — that's a segment, and I can measure its length with Pythagoras. One warning: if the two dots are actually the same dot, the arrow has zero length, the knob does nothing, and only a segment survives — as a single point of length zero.

---

See also: Vectors · Coordinate Geometry · Distance Formula · Midpoint Formula · Colinear Points · Angles · Parallel and Perpendicular Lines