Lines and planes in 3D — vector equations: Foundations

Below you will meet symbols like $a$ (a position), $d$ (a direction), and $r$ (the position of a general, roaming point). Do not worry that they are undefined yet — this whole note exists to build each one from nothing, in the order they depend on each other. By the last section you will be able to read $r = a + t d$ and $(r - a) \cdot n = 0$ with every mark earned. Skip nothing.

0. The scene: 3D space and the origin

Picture an empty room. Pick one corner and call it $O$ , the origin — the "you are here" dot that everything is measured from. From $O$ we draw three arrows at right angles: one along the floor to the right ( $x$ ), one along the floor away from you ( $y$ ), one straight up ( $z$ ). These are the axes.

We fix them in one standard arrangement called right-handed: point the fingers of your right hand along $x$ and curl them towards $y$ ; your thumb then points along $z$ . Every picture on this page — and the whole topic — uses this handedness. (If you flipped $z$ to point down instead, cross products would come out backwards, so we pin it down now.)

Any spot in the room is reached by "go $x$ steps right, $y$ steps forward, $z$ steps up." Those three numbers are the spot's coordinates, written $(x, y, z)$ .

Look at Figure s01: the red dot $P$ is reached by three dashed perpendicular steps from the corner $O$ — first along $x$ , then $y$ , then $z$ .

Figure — Lines and planes in 3D — vector equations

Figure s01 — one point pinned by three perpendicular steps from the origin in right-handed axes.

1. Vector — the "arrow" $a$ , $d$ , $r$

A vector is an arrow. It has a length (how far) and a direction (which way), but it does not care where it starts — slide it anywhere and it's the same vector. We write vectors in bold: $a$ , $d$ , $n$ , $r$ . On paper you may see $a$ instead; same thing.

A vector in 3D is stored as its three step-sizes: $d = (a, b, c)$ means " $a$ along $x$ , $b$ along $y$ , $c$ along $z$ ." These three numbers are its components.

Three names you will meet again and again:

$a$ — a fixed position vector (the arrow from $O$ to a chosen anchor point $A$ ).
$d$ — a direction vector (which way a line travels).
$r$ — the position vector of a general, roaming point: think of it as "the arrow from $O$ to whichever point we are currently talking about." As that point moves along a line or plane, $r$ changes; $a$ and $d$ stay put.

This is why the parent writes "point $A$ (position vector $a$ )": $A$ is the dot, $a$ is the arrow from $O$ that reaches it.

2. Adding, subtracting, and stretching — $+$ , $-$ , $t d$

Adding two vectors = "do one walk, then the other." Draw $a$ , then start $d$ where $a$ ended; the arrow from the very start to the very end is $a + d$ . In components you just add slot by slot: $a + d = (a_{1} + d_{1}, a_{2} + d_{2}, a_{3} + d_{3}) .$

Subtracting is just as important, because the topic constantly writes $r - a$ . Component-wise: $r - a = (r_{1} - a_{1}, r_{2} - a_{2}, r_{3} - a_{3}) .$ Its picture is the crucial one: $r - a$ is the arrow that goes from point $A$ to point $R$ (the "difference arrow," often written $A R$ ). Reason: to get from $O$ -to- $R$ , you could first go $O$ -to- $A$ (that's $a$ ) then $A$ -to- $R$ ; so $a + (r - a) = r$ , which is exactly why the leftover piece $r - a$ points from $A$ to $R$ . Whenever you see " $r - a$ ," picture the arrow joining the anchor to the roaming point.

Scaling a vector by a number $t$ (a plain number is called a scalar) = "keep the direction, change the length by the factor $t$ ." So $2 d$ is twice as long; $- 1 d$ flips it around; $0 d$ shrinks it to nothing. $t d = (t a, t b, t c) .$

Figure s02 shows both moves: on the left, the red arrow $r - a$ joining anchor $A$ to roaming point $R$ ; on the right, $d$ stretched to $2 d$ (red, longer) and flipped to $- d$ .

Figure s02 — left: the difference arrow $r - a$ runs from $A$ to $R$ ; right: scaling stretches or flips an arrow without rotating it.

3. Parallel vectors and "scalar multiple"

Two vectors are parallel when they point the same way or exactly opposite ways — i.e. one is just a stretched/flipped copy of the other. In symbols: $u$ is parallel to $v$ exactly when $u = t v$ for some scalar $t$ .

Why the topic needs it: the entire line derivation rests on one sentence — " $r - a$ (the arrow from anchor to roaming point, from §2) lies along the line, so it is parallel to $d$ ." Parallel means "scalar multiple," which gives $r - a = t d$ .

Recall Quick check

If $d = (2, 4, 0)$ , is $(1, 2, 0)$ parallel to it? ::: Yes — it equals $\frac{1}{2} d$ , a scalar multiple.

4. The parameter $t$ (and $s$ )

$t$ is the parameter: a free-running dial you may set to any real number, written $t \in R$ (" $t$ belongs to the real numbers"). Each value of $t$ hands you one point on the line. Turn the dial through all values and you trace the whole line.

A plane needs two independent dials, $s$ and $t$ , because you can slide in two independent directions across a flat surface.

See Parametrisation and parameters for the general idea of describing a shape by tunable inputs.

5. Length (magnitude) — $∣ v ∣$

The bars $∣ v ∣$ mean the length of the arrow $v$ . By Pythagoras in 3D: $∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2} .$ Picture the arrow as the long diagonal of a box whose sides are $v_{1}, v_{2}, v_{3}$ ; its length is that formula.

Why the topic needs it: to make a unit vector (length exactly 1) we divide by the length, $\hat{n} = n /∣ n ∣$ , provided $n \neq = 0$ (a zero-length arrow has no direction and $∣ n ∣ = 0$ , so we may not divide by it). The hat $^$ always means "length-1 version." Distances in the parent note all divide by $∣ n ∣$ or $∣ d ∣$ — legal precisely because a plane's normal and a line's direction are, by definition, nonzero.

6. The dot product — $u \cdot v$

The dot product takes two vectors and returns one number: $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} = ∣ u ∣ ∣ v ∣ cos θ,$ where $θ$ is the angle between them and $cos θ$ ("cosine") measures how aligned they are: $cos 0^{\circ} = 1$ (same direction), $cos 9 0^{\circ} = 0$ (perpendicular), $cos 18 0^{\circ} = - 1$ (opposite).

Why do the two formulas agree? (outline) Place $u$ and $v$ tail-to-tail. The straight line joining their tips is the vector $u - v$ , and its length obeys the cosine rule from triangle geometry: $∣ u - v ∣^{2} = ∣ u ∣^{2} + ∣ v ∣^{2} - 2∣ u ∣∣ v ∣ cos θ$ . Now expand the left side with the component rule (multiply out $(u - v) \cdot (u - v)$ slot by slot): everything cancels except $- 2 (u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3})$ . Matching the two expressions forces $u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} = ∣ u ∣∣ v ∣ cos θ$ . So the "multiply-and-add" recipe and the "angle" meaning are the same number wearing two costumes.

Why this particular tool? The topic constantly asks "is this arrow perpendicular to that one?" The dot product answers it with a single test: the dot product is zero exactly when the two vectors are at right angles (because $cos 9 0^{\circ} = 0$ ). That is the entire engine behind the plane's normal form $(r - a) \cdot n = 0$ . (The VERIFY block at the end of this page checks the perpendicular example below on real numbers.)

Figure s03 draws two perpendicular arrows $u$ and $v$ with the little right-angle square; their dot product is $0$ .

Figure s03 — a dot product of zero is exactly a right angle between the two arrows.

Full details live in Vectors and scalar (dot) product and its geometric use in Vector projection.

Recall Perpendicular test

$(1, 2, 2) \cdot (2, - 2, 1) = ?$ and are they perpendicular? ::: $2 - 4 + 2 = 0$ , so yes, perpendicular.

7. The cross product — $u \times v$

The cross product takes two vectors and returns a new vector that is perpendicular to both of them at once. Its components: $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1}) .$

Why does this formula give something perpendicular to both? (outline) Perpendicular means "dot product zero" (§6). So test it: dot $u \times v$ with $u$ and multiply everything out — the six terms cancel in pairs and you get exactly $0$ ; the same happens dotting with $v$ . (The VERIFY block at the end of this page checks exactly this on real numbers.) That built-in cancellation is why the topic uses this exact component recipe. Which of the two opposite perpendicular directions it points is fixed by the right-handed axes from §0.

The parallel edge-case (this is why a plane needs non-parallel spanning arrows). The length of $u \times v$ equals $∣ u ∣ ∣ v ∣ sin θ$ . If $u$ and $v$ are parallel (or one is zero), then $θ = 0^{\circ}$ (or $18 0^{\circ}$ ), so $sin θ = 0$ and the cross product collapses to the zero vector $0$ — which points nowhere and is useless as a normal. That is precisely why the parent insists a plane's two spanning directions $u, v$ be non-parallel: only then does $u \times v$ produce a genuine, nonzero normal.

Why this particular tool? A plane is spanned by two in-plane directions $u, v$ . Its normal must stick out perpendicular to both. No single dot product gives that — but the cross product is designed to output the one arrow orthogonal to two inputs. That is why the parent writes $n = u \times v$ .

Figure s04 shows two arrows lying in a shaded plane and the red cross product standing straight up out of it.

Figure s04 — the cross product (red) stands perpendicular to both arrows; it vanishes to zero if they are parallel.

8. Normal vector — $n$

The normal $n$ of a plane is any arrow pointing straight out of the plane, at right angles to every direction lying inside it. It is not along the plane — it is across it. This is the opposite role to a line's direction $d$ , which lies along the line.

The prerequisite map

The diagram below is a dependency chart: read it top to bottom. Each box is one idea from this note, and an arrow " $A \to B$ " means "you need $A$ before $B$ makes sense." Coordinates make vectors; vectors get added/subtracted/scaled/measured; those feed the two special products; and everything funnels down into the parent topic. (If the diagram fails to render in your reader, the same order is exactly §0 → §8 of this page.)

Everything converges on the parent topic.

Equipment checklist

Cover the answers and test yourself — if any one is shaky, reread its section before the parent note.

A vector stores three numbers meaning… ::: how far to step along each of the $x, y, z$ axes.
The difference between a point and a position vector is… ::: a point is a fixed spot; a position vector is the arrow from the origin $O$ reaching that spot (same three numbers).
$r - a$ is the arrow that… ::: goes from anchor point $A$ to the roaming point $R$ (the difference arrow).
$t d$ geometrically does what to the arrow $d$ … ::: stretches/shrinks it by factor $t$ (and flips it if $t < 0$ ), keeping direction.
"Two vectors are parallel" means in symbols… ::: one equals a scalar multiple of the other, $u = t v$ .
Is the zero vector a valid direction for a line? ::: No — it has no direction; a line's $d$ must satisfy $d \neq = 0$ .
The parameter $t \in R$ contributes what to a line… ::: each value of $t$ picks one point; all values sweep the whole line.
$∣ v ∣$ is computed as… ::: $v_{1}^{2} + v_{2}^{2} + v_{3}^{2}$ , the arrow's length by Pythagoras.
A unit vector $\hat{n}$ is… ::: $n$ divided by its length $∣ n ∣$ (requires $n \neq = 0$ ), giving length exactly 1.
The dot product $u \cdot v$ equals zero exactly when… ::: the two vectors are perpendicular ( $9 0^{\circ}$ ).
The cross product $u \times v$ outputs a vector that is… ::: perpendicular to both $u$ and $v$ (side fixed by the right-hand rule).
What is $u \times v$ when $u$ and $v$ are parallel? ::: the zero vector $0$ (since $sin θ = 0$ ) — which is why plane spanning directions must be non-parallel.
A plane's normal $n$ points… ::: straight out of (across) the plane, unlike a line's direction which lies along it.

Connections

Vectors and scalar (dot) product — the dot product built here powers the plane's normal form.
Cross product — the perpendicular-to-both tool that makes normals.
Vector projection — uses the dot product and unit vectors for distances.
Parametrisation and parameters — the meaning of the dials $t$ and $s$ .
Systems of linear equations — where these symbols get solved together.
Intersection of lines and planes — the payoff once the symbols are fluent.

#flashcards/maths