Foundations — Distance formula in 3D

This is the tool-shed page for Distance formula in 3D. Every symbol the parent note uses is unpacked here, in the order that each one leans on the one before it. If a line in the parent felt like it skipped a step, the missing step lives here.

0 · What a point in space even means

Look at the figure below. The point is pinned at the corner of a little box built out of three measuring sticks meeting at a shared corner — the origin.

• The number $x$ tells you how far to slide along the x-axis (right).
• The number $y$ tells you how far to slide along the y-axis (into the page).
• The number $z$ tells you how far to climb along the z-axis (up).

Why the topic needs it: the distance formula compares two such points, so we must first be able to pin one point down unambiguously. The three-number address is what makes that possible.

The origin is the special point $O(0,0,0)$ — the shared corner where all three sticks start. See Coordinate axes and octants in 3D for how these three axes carve space into eight regions.

1 · The subscript notation $x_1,x_2$

Why the topic needs it: the formula subtracts same-named coordinates ($x_2-x_1$). Without subscripts we could not even write "the x of Q minus the x of P".

2 · A coordinate difference $x_2-x_1$

In the figure the three differences $x_2-x_1$, $y_2-y_1$, $z_2-z_1$ are the three edges of the box that has $P$ and $Q$ at opposite corners. That is the whole reason the parent note says "drop into a box" — each difference is one edge.

Why the topic needs it: these three gaps are the raw ingredients. Everything downstream is built from squaring and adding them.

3 · The absolute value $|\cdot |$ — and why we can drop it

The parent writes the box edges as $a=|x_2-x_1|$ etc. — with bars — because an edge length can never be negative. But then in the final formula the bars vanish. Why?

Why the topic needs it: this is exactly why the parent's third "common mistake" is true — order of subtraction does not matter. $x_2-x_1$ and $x_1-x_2$ differ only by a sign, and squaring kills that sign.

4 · Squaring $({)}^2$ and summing

Why does Pythagoras care about areas? Because the theorem is a statement about areas of squares, not lengths — see the next section.

Why the topic needs it: the formula sums three squared differences. Picturing each as a tile-area makes the addition meaningful rather than a mystery.

5 · The square root ​

Why the topic needs it: squaring and adding gives us the hypotenuse squared; the root brings us back to an actual length.

6 · Pythagoras' theorem — the engine

Read it in area-language (section 4): the tile on the long side has area equal to the two smaller tiles combined. That is why squares appear — Pythagoras is fundamentally about the balance of square areas.

Why the topic needs it: the 3D formula is Pythagoras applied twice — once for the floor diagonal, once tilting up. Master this triangle and the derivation is just repetition. See Distance formula in 2D for the single-Pythagoras version.

7 · The vector $\overrightarrow{PQ}$ and its magnitude

So "distance $PQ$" and "magnitude of $\overrightarrow{PQ}$" are two names for the same number. The parent's key-results box states this; here you see it is not a coincidence — the arrow's components are the box edges of section 2.

Why the topic needs it: it links this whole chapter to Vectors and magnitude and to Direction cosines and direction ratios, which are built from these same three differences.

Prerequisite map

Equipment checklist

Self-test: can you answer each before revealing?

Connections

• Distance formula in 3D — the topic these foundations feed.
• Coordinate axes and octants in 3D — where the three-number address comes from.
• Distance formula in 2D — Pythagoras used once.
• Vectors and magnitude — distance as an arrow's length.
• Direction cosines and direction ratios — also built from the differences.
• Section formula in 3D — same coordinate machinery.
• Equation of a sphere — "all points at fixed distance".