Foundations — Section formula in 3D

Before you can even read the parent note, a whole crowd of little symbols shows up: $A$, $B$, $x_1$, $\vec{a}$, $m:n$, arrows over letters, subscripts. This page hunts down every single one and builds it from nothing. If a symbol appears in the parent, it is defined here first — with a picture.

We build them in an order where each one leans on the one before it.

1. A point, and its three numbers $(x,y,z)$

Why three? Because space has three independent directions. Stand in a room: I can tell you left–right, forward–back, and up–down. Three answers locate you exactly — no more, no less.

We write the point as an ordered list: $P=(x,y,z).$

• $x$ = the right/left number (first axis).
• $y$ = the forward/back number (second axis).
• $z$ = the up/down number (third axis).

The order matters: $(1,2,3)$ is a different spot from $(3,2,1)$. That is why we call it an ordered triple.

2. Subscripts: $x_1$, $x_2$, $y_1$ … telling two points apart

The section formula juggles two points at once, $A$ and $B$. Both have an $x$. How do we avoid confusion?

So we write: $A=(x_1,y_1,z_1),B=(x_2,y_2,z_2).$

Read $x_1$ out loud as "ex-one", never "ex times one".

3. The segment $AB$ and a point $P$ living on it

A point $P$ on the segment is a spot somewhere along that rope. As $P$ slides from $A$ toward $B$, its three coordinates change smoothly. Our whole job is: given how far along $P$ sits, what are its three numbers?

This picture is the beating heart of the topic. Everything else is bookkeeping. See Distance formula in 3D if you want to measure the actual lengths $AP$ and $PB$.

4. The ratio $m:n$ — how the rope is split

Now, where exactly on the rope is $P$? We describe it not by a length but by a ratio.

$\frac{AP}{PB}=\frac{m}{n}.$

• If $m:n=1:1$, the two pieces are equal → $P$ is the middle.
• If $m:n=2:3$, the $A$-side piece has 2 parts, the $B$-side piece has 3 parts, so $P$ sits closer to $A$ (its side is the shorter one).

5. Vectors: the arrow $\vec{a}$ and the position vector

The parent note gives a slick "vector method". Let us earn that arrow.

We will also need the position vector of the moving point $P$: it is written $\vec{p}$, the arrow from $O$ to $P$, carrying $P$'s three (as yet unknown) numbers: $\vec{p}=(x,y,z).$ Finding $\vec{p}$ is the whole problem — once we know this arrow, we read off $x,y,z$.

Why bother, if these are the same numbers as the point? Because arrows can be subtracted to get the arrow between two points — and that is the move the derivation needs.

Why subtraction? Look at the figure: to walk from $A$ to $B$ you first "undo" the trip out to $A$ (that's $-\vec{a}$) and then take the trip out to $B$ (that's $+\vec{b}$). The origin cancels, leaving the pure $A\to B$ arrow. Deepen this in Position vectors.

Using the same rule for our unknown $P$: $\overrightarrow{AP}=\vec{p}-\vec{a},\overrightarrow{PB}=\vec{b}-\vec{p}.$ These two arrows are the two rope-pieces $AP$ and $PB$, now written as arrows.

6. Scaling an arrow: multiplying a vector by a number

The parent's key step, $\overrightarrow{AP}=\frac{m}{n}\overrightarrow{PB}$, multiplies an arrow by a number. We have not yet said what that means. Fix it now.

Now the vector-ratio statement makes sense. Saying $\overrightarrow{AP}=\frac{m}{n}\overrightarrow{PB}$ packs two facts into one arrow equation:

1. Lengths: $\frac{AP}{PB}=\frac{m}{n}$ — the rope-split ratio from §4, now as arrow lengths.
2. Direction: for internal division $\overrightarrow{AP}$ and $\overrightarrow{PB}$ point the same way (both run "toward $B$"), so the scalar $\frac{m}{n}$ is positive — no flip.

That is what "a ratio of arrows" means: same-line, length in proportion $m:n$, sign carrying the direction.

7. The derivation, one honest step at a time

We now build $x=\frac{m{x}_2+n{x}_1}{m+n}$ from scratch, doing the algebra fully.

Step 1 — write the ratio as an arrow equation. $\overrightarrow{AP}=\frac{m}{n}\overrightarrow{PB}.$ What: stated the rope-split $AP:PB=m:n$ using §6's scaling. Why: it links the two pieces with one clean equation. Looks like: two arrows on the same line, the $A$-side one being $\frac{m}{n}$ as long.

Step 2 — replace the arrows by position vectors (§5): $\vec{p}-\vec{a}=\frac{m}{n}(\vec{b}-\vec{p}).$

Step 3 — clear the fraction by multiplying both sides by $n$: $n(\vec{p}-\vec{a})=m(\vec{b}-\vec{p}).$

Step 4 — open the brackets (scalar multiplication distributes over each entry, §6): $n\vec{p}-n\vec{a}=m\vec{b}-m\vec{p}.$

Step 5 — gather all $\vec{p}$ on one side: $n\vec{p}+m\vec{p}=m\vec{b}+n\vec{a}⟹(m+n)\vec{p}=m\vec{b}+n\vec{a}.$

Step 6 — divide by the number $m+n$: $\vec{p}=\frac{m\vec{b}+n\vec{a}}{m+n}.$

Step 7 — read off one coordinate. Because §1 said each axis is independent, the $x$-entry of $\vec{p}$ uses only the $x$-entries of $\vec{a},\vec{b}$: $x=\frac{m{x}_2+n{x}_1}{m+n}.$

There it is — earned, not decoded. The identical algebra on the second and third entries gives $y$ and $z$ with the same $m,n$.

8. Edge cases: what if $m$ or $n$ is $0$ or negative?

The formula quietly handles the corners. Check them so nothing surprises you.

Prerequisite map

Every arrow feeding into Section formula in 3D is a symbol this page just built. Follow them onward to Equation of a line in 3D, where the same ratio idea becomes a moving parameter, and to Centroid of a triangle in 3D, which is one clever ratio of $2:1$.

Equipment checklist

Test yourself — reveal only after you have answered aloud.