Foundations — Position vector, displacement, distance

This page assumes nothing. Every letter, hat, arrow, and squiggle used in the parent note is built here from the ground up, in an order where each idea leans only on the ones before it.

0. The very first thing: a point and a dot to measure from

Before any symbols, look at a flat sheet of paper. Put one thing on it — say a cat. Now ask: "where is the cat?"

You cannot answer without a second mark: a fixed reference dot everyone agrees on. We call that dot the origin and label it $O$.

WHY we need it: the number "3" is meaningless. "3 metres to the right of $O$" is a real place. Location only exists relative to a reference. See Coordinate systems and unit vectors.

1. Axes, coordinates, and the four quadrants

From $O$ we lay down two rulers at right angles: one going right (the x-axis), one going up (the y-axis). A third, coming out of the page toward you, is the z-axis — we mostly ignore it on flat problems.

Crucially, each ruler runs both ways. Stepping right is positive $x$; stepping left is negative $x$. Stepping up is positive $y$; stepping down is negative $y$. So a coordinate is a signed number: the sign says which way along the ruler.

The two axes chop the flat plane into four quadrants, named I–IV by the signs of $(x,y)$: I is $(+,+)$ top-right, II is $(-,+)$ top-left, III is $(-,-)$ bottom-left, IV is $(+,-)$ bottom-right.

WHY signs matter: motion goes every direction, not just right-and-up. Without negatives you could never describe a point to the left of $O$ or below it.

WHY fix it: without a stated orientation, "out of the page" is ambiguous — someone could flip the z-axis. The right-hand rule pins it down.

WHY coordinates at all: numbers are easier to calculate with than pictures. Coordinates are the bridge from geometry to arithmetic.

2. What a "vector" even is — an arrow with a length and a direction

Here is the star of the whole chapter.

Contrast this with a scalar — just a plain number with no direction (like temperature, or a length). This distinction is the spine of Scalars and vectors.

The little hat symbol $\hat{}$ is coming next — hold on.

3. Unit vectors $\hat{i},\hat{j},\hat{k}$ — the "one step" arrows

To build any arrow out of numbers, we need standard "one-step" arrows to build with.

WHY these: any arrow on the plane can be reached by "so many $\hat{i}$'s to the right, so many $\hat{j}$'s up." A negative number of steps just means the opposite direction — e.g. $-2\hat{i}$ is two steps left. So writing a generic arrow $\vec{v}$ as $\vec{v}=x\hat{i}+y\hat{j}+z\hat{k}$ literally means: "start at $O$, take $x$ steps along x, then $y$ along y, then $z$ along z," where negative components reverse the direction. The signed coordinates from §1 have become the components of the arrow. More in Coordinate systems and unit vectors.

4. The position vector $\vec{r}$ — the arrow to "here"

Now we can name the topic's first big object. From here on, the specific symbol $\vec{r}$ is reserved for it.

The symbol $\vec{r}$ is just a name — "$r$" for radius/reach from the origin, with the vector-arrow on top so you never forget it has direction.

WHY it matters: the zero vector is the one case where "which way does the arrow point?" has no answer — and it is exactly the displacement of any round trip that returns to its start.

5. Magnitude $|\vec{r}|$ and the Pythagoras engine

Often we want only the length of an arrow — how far the point is from $O$ — throwing away direction.

How do we get a length from components? The components $x$ (across) and $y$ (up) are the two legs of a right triangle, and the arrow $\vec{r}$ is its slanted side (hypotenuse). Because the legs get squared, any negative signs vanish — length comes out positive whatever quadrant P is in.

WHY this tool and not another? We have the two perpendicular legs and want the slanted length. The only rule that connects the two legs of a right triangle to its slant is Pythagoras theorem — "the square of the slant equals the sum of the squares of the legs." That is precisely the question we're asking, so that is the tool we reach for. The square root simply undoes the "squared" to return a plain length.

6. The symbol $\Delta$ — "change in"

The little letters $i$ and $f$ tacked on (${\vec{r}}_i$, ${\vec{r}}_f$) mean initial (start) and final (end). So $\Delta \vec{r}={\vec{r}}_f-{\vec{r}}_i$ is read "the change in position = end arrow minus start arrow."

WHY we need it: motion is about change, not absolute location. $\Delta$ is the compact way to say "how much did this shift?"

7. Subtracting arrows — the picture behind ${\vec{r}}_f-{\vec{r}}_i$

The whole idea of displacement rests on one operation: subtracting two arrows. Here is what it looks like.

${\vec{r}}_f-{\vec{r}}_i=(x_f-x_i)\hat{i}+(y_f-y_i)\hat{j}$

This is the machinery of Vector addition and subtraction. Notice the origin $O$ shows up inside both ${\vec{r}}_i$ and ${\vec{r}}_f$ — so when you subtract, it cancels. That is why displacement doesn't care where you put $O$. If start and end coincide, this subtraction gives $\vec{0}$ — the zero vector from §4.

8. Summation $\sum$ — adding up the road

For distance we add the lengths of every piece of the path.

WHY: the path can bend any number of times; $\sum$ is the tidy way to say "keep adding, however many pieces there are." Each piece's length comes from Pythagoras (§5) again.

9. The inequality symbol $\le$

WHY we need it (and not just $<$ or $=$): the topic's headline result compares the straight gap to the walked road, and these two are sometimes equal (a straight, one-way walk) but usually the road is longer (any bend or reversal). A single symbol must cover both cases at once — that is exactly what $\le$ does. Using plain $<$ would falsely forbid the straight-line tie; using $=$ would falsely forbid every bent path.

The topic's headline result $|\Delta \vec{r}|\le \text{distance}$ reads: "the straight-line gap is never longer than the road you walked — at best they tie." They tie only when you never turn or double back — the special case studied in Motion in a straight line.

10. Putting the vocabulary to work (mini example)

This ties directly into Average velocity and average speed: velocity is built from $\Delta \vec{r}$, speed from distance.

Equipment checklist

Test yourself — say the answer out loud before revealing.