Foundations — Equation of a plane — normal form, intercept form, general form

This page assumes you have seen none of the notation the parent note throws around. We build each symbol from a picture, in an order where every step leans only on the step before it. By the end you will be able to read $ax+by+cz+d=0$ the way you read a road sign.

0. What is a point in 3D? — the ordered triple $(x,y,z)$

Before planes, before arrows, we need to say where things are.

Look at Figure s01: the three coloured segments show the walk — magenta $x=3$ along the first axis, violet $y=2$ into the room, orange $z=4$ straight up — landing on the point $P$. The figure is the definition made visible: a point is nothing but a recipe of three walks.

Why we need this: a plane is a set of points. To talk about "all the points on a plane" we must first be able to name a single point. The triple $(x,y,z)$ is that name.

The letter names are pure convention — $x$ first, $y$ second, $z$ third. The number zero in a slot means "don't move along that axis": $(3,0,0)$ sits on the $x$-axis alone.

1. What is a vector? — the arrow $\vec{v}$

The picture: put the tail of the arrow at the origin; its head lands at the point $(a,b,c)$. So a vector and a point share the same three numbers — the difference is interpretation. A point is a place; a vector is a displacement (a move from somewhere to somewhere).

2. Scaling a vector — multiply an arrow by a plain number

Before we can subtract or shrink arrows, we need one small operation: multiplying a vector by an ordinary number (a scalar).

Why the topic needs it: every "make it a unit vector" step ($\hat{n}=\vec{n}/|\vec{n}|$) and every "divide the equation by $|\vec{n}|$" in the parent note is exactly this operation — squeezing an arrow's length while keeping its direction. Get this now and those later steps stop looking mysterious.

3. Subtracting positions — the arrow between two points

If ${\vec{r}}_1$ and ${\vec{r}}_2$ are two points, what is ${\vec{r}}_2-{\vec{r}}_1$?

Look at Figure s02: the violet and orange arrows are the two position vectors ${\vec{r}}_1,{\vec{r}}_2$ drawn from the origin; the bold magenta arrow is their difference ${\vec{r}}_2-{\vec{r}}_1$, sitting with its tail on ${\vec{r}}_1$ and head on ${\vec{r}}_2$. Notice it does not start at the origin — that is the whole point of a difference arrow: it connects two places.

Why the topic needs this: the parent note constantly writes things like $\vec{n}\cdot (\vec{r}-{\vec{r}}_0)=0$. Here ${\vec{r}}_0$ is a specific, known point that lies on the plane (think of it as the one pin we've already stuck in the cardboard), while $\vec{r}$ is any mystery point we are testing. Then $\vec{r}-{\vec{r}}_0$ is the arrow lying inside the plane, running from the known pin ${\vec{r}}_0$ to the mystery point $\vec{r}$. You cannot understand "perpendicular to the plane" until you can see that in-plane arrow — and it is born from subtraction.

4. Length of a vector — the magnitude $|\vec{v}|$

How long is the arrow $(a,b,c)$?

Why the topic needs it: to convert a normal into a unit normal (length exactly 1), you divide by its length. The parent's $\sqrt{a^2+b^2+c^2}$ that keeps appearing is exactly this length — it is not a mysterious constant, it is just "how long is the normal."

5. Unit vector — the hat $\hat{n}$

6. Direction cosines — the numbers $(l,m,n)$

The three components of a unit normal get a special name.

Why the topic needs it: the parent's normal form $lx+my+nz=p$ uses exactly these. The condition $l^2+m^2+n^2=1$ is just "$\hat{n}$ has length 1" wearing a costume. (Full details live in Direction cosines and direction ratios.)

7. The dot product — the shadow $\vec{u}\cdot \vec{v}$

This is the engine of the whole topic. Read this section twice.

Look at Figure s03: the dashed violet line drops $\vec{v}$ straight down onto the line of $\vec{u}$; the thick orange arrow is the resulting shadow. The dot product measures the length of that orange shadow (times $|\vec{u}|$). As you swing $\vec{v}$ towards a right angle with $\vec{u}$, the shadow shrinks — and vanishes exactly when they are perpendicular.

Why the topic needs it: "normal to the plane" means "perpendicular to every arrow lying in the plane." Turn that into arithmetic and you get $\vec{n}\cdot (\vec{r}-{\vec{r}}_0)=0$ — the point–normal form. No dot product, no plane equation. (Deeper: Dot product and projection.)

One more fact the parent leans on: a vector dotted with itself: $\hat{n}\cdot \hat{n}=|\hat{n}{|}^2\cos 0^{\circ }=|\hat{n}{|}^2=1(\text{since }\hat{n}\text{ has length }1).$ That is why the $p(\hat{n}\cdot \hat{n})$ term in the parent's normal-form derivation collapses to just $p$.

8. The cross product — the built-in normal $\vec{u}\times \vec{v}$

Sometimes you don't know the normal; you only know three points on the plane. The cross product hands you a normal for free.

Here $\hat{i},\hat{j},\hat{k}$ are the three standard unit vectors along the axes: $\hat{i}=(1,0,0)$, $\hat{j}=(0,1,0)$, $\hat{k}=(0,0,1)$. They are just labelled names for "one step along each axis."

Why the topic needs it: if $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are two arrows lying in the plane (and not parallel), then $\overrightarrow{AB}\times \overrightarrow{AC}$ pokes straight out of the plane — a ready-made normal. That is the entire trick behind "plane from three points." (More: Cross product as area/normal.)

9. The equals-zero equation — what "the equation of a plane" even means

Look at Figure s04: the navy line is the set of all points that make the expression equal zero. The violet point $(1,2)$ scores $0$ — it sits on the line (ON). The magenta point $(3,3)$ scores $5\ne 0$ — it is off the line (OFF). Same test, two verdicts. A plane in 3D works identically; the picture is a 2D slice so you can see it.

Prerequisite map

The chain reads: name points, turn them into arrows, learn to scale and subtract them (in-plane arrow), measure length (unit normal, direction cosines), test perpendicularity (dot product), manufacture a normal (cross product) — and all of it converges on the plane equation.

Equipment checklist

Test yourself — reveal only after you have answered aloud.

Connections

• Equation of a plane — normal form, intercept form, general form (parent)
• Direction cosines and direction ratios
• Dot product and projection
• Cross product as area/normal
• Distance of a point from a plane
• Vector and Cartesian equation of a line
• Angle between two planes