Foundations — Distance from a point to a plane

This page assumes you have seen nothing. Before we can even read the sentence "distance from $P$ to plane $ax+by+cz+d=0$," we must earn every letter in it. We build the pieces bottom-up, each one leaning only on the ones before it.

0. Points and coordinates — where "$(x,y,z)$" comes from

Look at the figure: from the origin, the butter dashed steps go $x$ along the floor, then $y$ across the floor, then $z$ straight up. The coral dot at the end is the point $P$. The little symbols mean:

• $x_1,y_1,z_1$ — the three coordinates of our point $P$. The subscript $1$ just says "these belong to point number 1," to keep them apart from another point's coordinates.
• $(x_0,y_0,z_0)$ — later we'll use subscript $0$ for a point sitting on the plane. Same idea, different label.

Why the topic needs this: distance is a number that comes out of the coordinates of a point and the numbers describing a plane. No coordinates, nothing to compute.

1. Vectors — an arrow with a length and a direction

The difference between a point and a vector: a point is a place; a vector is a trip. If you subtract two points you get the trip between them:

$\overrightarrow{QP}=(x_1-x_0,y_1-y_0,z_1-z_0).$

Why the topic needs this: the whole derivation measures the arrow from the wall to the point, then asks how much of that arrow points straight at the wall.

2. Length (magnitude) — the size of an arrow

The figure shows why the square-root-of-squares works: it's Pythagoras done twice. First find the diagonal across the floor ($\sqrt{v_x^2+v_y^2}$, the mint segment), then combine that with the vertical rise ($v_z$, the lavender segment) to get the true 3D length (the coral arrow).

Why the topic needs this: length is how we turn a "raw scaled number" into an honest distance. Dividing by $|\vec{n}|$ is what rescales.

3. The dot product — "how much do two arrows agree?"

There is a second face of the same formula: $\vec{u}\cdot \vec{v}=|\vec{u}||\vec{v}|\cos \theta ,$ where $\theta$ is the angle between the arrows. We use $\cos$ (not $\sin$ or $\tan$) here because cosine is the function that reads off "how aligned" — it is $1$ when arrows coincide ($\theta =0$), $0$ when they're perpendicular ($\theta =90^{\circ }$), and $-1$ when opposite. That is exactly the "agreement" behaviour we want.

Why the topic needs this: projecting the arrow onto the normal (the heart of the derivation) is a dot product.

4. The unit vector — a direction stripped of length

5. Projection — the shadow of one arrow on another

Imagine sunlight shining perpendicular to $\hat{n}$. The coral arrow $\vec{u}$ casts a lavender shadow onto the $\hat{n}$ line. The length of that shadow (the thick lavender segment) is the projection. The rest of $\vec{u}$ — the part running sideways — contributes nothing to the shadow, which is why only the perpendicular component survives.

Why the topic needs this: the signed distance $D=\overrightarrow{QP}\cdot \hat{n}$ is literally this shadow.

6. The plane and its equation — the wall, and its secret recipe

The figure shows the flat mint plane and its coral normal arrow $\vec{n}=(a,b,c)$ poking straight out of it. Notice:

• $a,b,c$ — the coefficients. Together they form the normal vector $\vec{n}=(a,b,c)$: the arrow the plane "faces" along, perpendicular to the sheet. See Normal vector to a plane and Equation of a plane.
• $d$ — a constant that shifts the plane toward or away from the origin. With $d=0$ the plane passes through the origin; changing $d$ slides it parallel to itself.
• $=0$ — the equation is a test: plug a point's coordinates in; if you get exactly $0$, the point is on the plane. Get a non-zero number, and (after rescaling) that number tells you how far off you are.

7. Signed vs unsigned — sign for side, bars for size

Distance can never be negative, so the final formula wears bars. But before stripping the sign, the sign itself is information:

• positive ⇒ the point is on the side $\vec{n}$ points toward,
• negative ⇒ the opposite side,
• zero ⇒ on the plane.

Why the topic needs both: $|\cdot |$ for the honest distance; the bare sign for "which side?" questions and same-side tests.

How the pieces feed the topic

Read top to bottom: coordinates make vectors; vectors have length and dot products; those two make a unit normal; the unit normal lets us take a projection; the projection plus the length-division plus the absolute value give the final distance.

Equipment checklist

Hide the right side and test yourself — if any line stumps you, reread its section above.

Connections

• Equation of a plane — the source of $(a,b,c,d)$.
• Normal vector to a plane — why $\vec{n}=(a,b,c)$ points straight out.
• Projection of a vector — the shadow/dot-product step.
• Parent: Distance from a point to a plane — where all this equipment is used.
• Distance between two parallel planes — a direct sequel.
• Distance from a point to a line in 3D — the sister problem.
• Foot of perpendicular from a point to a plane — landing the trip on the plane.