Quaternion product — Hamilton product: Foundations

This page assumes nothing. Before you can read "multiply two quaternions," you must own every symbol that appears: what a scalar is, what a vector is, what an axis and an angle are, what a dot product and a cross product measure, and what those strange letters $i, j, k$ actually do. We build them one at a time, each on top of the last.

0. Numbers you already trust — the scalar

WHY we need it: a quaternion has one scalar slot (called $w$ ) that stores "how much of the rotation is no turn at all." When two rotations point in similar directions, the scalar part grows. It is the anchor everything else hangs off.

1. An arrow with direction — the vector

Figure — Quaternion product — Hamilton product

WHAT the picture shows: three mutually perpendicular chalk axes ( $x$ right, $y$ up-into-page, $z$ up), and a blue arrow $v$ whose shadow onto each axis gives its three numbers. Those shadows are the components $x, y, z$ .

WHY the topic needs it: a quaternion has three vector slots $(x, y, z)$ that together store the axis the rotation twists around. The scalar $w$ plus the vector $v$ is the whole quaternion: $q = (w, v)$ .

2. Axis and angle — how we describe a turn

WHAT the picture shows: a yellow axis arrow $\overset{n}{^}$ and a pink curved arrow sweeping the turn angle $θ$ around it. Point your right thumb along $\overset{n}{^}$ ; your curling fingers show the positive turn direction.

WHY the topic needs it: the parent's Example 2 builds a quaternion straight from an axis and angle via $q = (cos \frac{θ}{2}, sin \frac{θ}{2} \overset{n}{^})$ . You cannot read that line until axis and angle are solid. (Where the half-angle comes from is the job of Axis-Angle & Euler Rodrigues.)

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

WHY this tool and not another? The dot product answers exactly one question: "how aligned are these two arrows?" It is largest when they point the same way ( $cos 0 = 1$ ), zero when they are perpendicular ( $cos 9 0^{\circ} = 0$ ), and negative when they oppose. That perpendicular-gives-zero fact is why, in the parent's Worked Example 1, $v_{1} \cdot v_{2} = 0$ for $i$ and $j$ .

WHY the topic needs it: the scalar part of every Hamilton product is $w_{1} w_{2} - v_{1} \cdot v_{2}$ . That minus-dot is a direct measurement of axis alignment.

4. The cross product — "twist that points sideways"

WHAT the picture shows: two chalk arrows $a$ (blue) and $b$ (pink) lying flat, and their cross product $a \times b$ (yellow) standing straight up out of the board. Point right-hand fingers from $a$ toward $b$ ; the thumb is $a \times b$ .

WHY this tool and not another? We need something that captures "twist across" — a turning tendency that has its own direction. Only the cross product does that in 3D. It is why the vector part of the Hamilton product carries a $v_{1} \times v_{2}$ term.

5. The imaginary units $i, j, k$

6. Norm, conjugate, inverse — keeping a quaternion "valid"

WHY the topic needs it: rotating a vector uses the "sandwich" $q u q^{- 1}$ — you cannot read that until you know what $q^{- 1}$ and $\overset{q}{ˉ}$ mean. And the "always renormalize" mistake on the parent page only makes sense once you know $∣ q ∣$ .

Prerequisite map

Read it top-down: plain numbers and arrows build the quaternion; dot and cross build the two halves of the product; the units $i, j, k$ carry the cross-product's right-hand rule into the algebra; and norm/conjugate/inverse let you use a quaternion to rotate.

Equipment checklist

Cover the right side and test yourself.

A scalar carries this and only this

size (magnitude), no direction

A vector is drawn as

an arrow with a length and a direction, written

(x, y, z)

A unit vector (hat, like

\overset{n}{^}

) has length

exactly

1

To describe a turn you need which two things

an axis

\overset{n}{^}

and an angle

θ

a \cdot b

equals zero when the arrows are

perpendicular (at

9 0^{\circ}

)

The dot product cares about order?

No — it is symmetric,

a \cdot b = b \cdot a

a \times b

points

perpendicular to both, by the right-hand rule

The cross product cares about order?

Yes —

a \times b = - (b \times a)

Which ingredient makes the Hamilton product non-commutative

the cross product term only

i^{2} = j^{2} = k^{2}

equals

- 1

ij

equals

k

; and

j i

equals

- k

A unit quaternion has norm

∣ q ∣ = 1

The conjugate

\overset{q}{ˉ}

(w, v)

(w, - v)

For a unit quaternion

q^{- 1}

equals

\overset{q}{ˉ}

Connections

Parent: Hamilton product — where all these symbols get multiplied together.
Cross Product & Right-Hand Rule — the geometric root of $ij = k$ and of non-commutativity.
Axis-Angle & Euler Rodrigues — where axis, angle and the half-angle enter the quaternion.
Rotation Matrices — SO(3) — the same rotations written as $3 \times 3$ matrices.
Quaternion Kinematics — $\dot q = \tfrac12 q\,\omega$ — uses norm and the product every gyro tick.
Multiplicative EKF (MEKF) — attitude filters that chain these products.
Gimbal Lock & Euler Angles — the problem quaternions were built to dodge.