Quaternion product — Hamilton product: Visual walkthrough

Before we touch the formula, let us agree on what every symbol is, because we will not use one before you can picture it.

Step 0 — The four numbers and the three arrows

WHAT the picture shows: the three tags $i, j, k$ drawn as three arrows at right angles, exactly like the $x, y, z$ axes of a room. A quaternion's vector part $(x, y, z)$ is a single arrow built from these.

WHY we draw them this way: the entire multiplication rule comes from how these three arrows spin into each other. If you can see the room, you can read off every product.

The two facts we are allowed to use — nothing else:

Step 1 — The cyclic ring: why $ij = k$ and $j i = - k$

WHAT: the green arrows follow $i \to j \to k \to i$ (clockwise) and each hop is a product equal to the next letter: $ij = k$ , $j k = i$ , $k i = j$ . The red arrows go the other way and pick up a minus sign.

WHY this matters more than it looks: this is exactly the right-hand rule of the Cross Product & Right-Hand Rule. If $i, j, k$ were the unit vectors $\overset{x}{^}, \overset{y}{^}, \overset{z}{^}$ , then $\overset{x}{^} \times \overset{y}{^} = \overset{z}{^}$ — the same clockwise ring. So the "twist" half of quaternion multiplication is a cross product. Hold that thought; we prove it in Step 5.

PICTURE takeaway: swapping the order of two neighbours reverses the arrow, which flips the sign. That single fact — $ij \neq = j i$ — is the seed of non-commutativity, and non-commutativity is the seed of 3D rotation.

Step 2 — FOIL: turn one product into sixteen

WHAT the picture shows: a $4 \times 4$ grid. Rows are the parts of $p$ (the tags $1, i, j, k$ ), columns are the parts of $q$ . Each cell is one of the 16 products, coloured by where it eventually lands:

yellow cells → land in the scalar part,
blue cells → land in the $i$ / $j$ / $k$ vector part.

WHY a grid: the whole derivation is just sorting these 16 cells into two buckets — the scalar bucket and the vector bucket. No new idea is needed; we only apply the Step-1 rules to each cell and add up.

The diagonal cells are the dangerous ones: $x_{1} x_{2} = - 1 (ii), y_{1} y_{2} = - 1 (j j), z_{1} z_{2} = - 1 (k k)$ Each squares a tag to $- 1$ , so each falls out of the vector bucket and into the scalar bucket carrying a minus sign. That minus is the whole reason a dot product appears next.

Step 3 — Collect the scalar bucket → a dot product appears

WHAT: the picture shows two arrows $v_{1}, v_{2}$ and shades the quantity $v_{1} \cdot v_{2} = ∣ v_{1} ∣∣ v_{2} ∣ cos α$ — the amount they point the same way.

WHY the dot product, and not some other combination? We did not choose it. The three diagonal $- 1$ 's forced the sum $x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}$ , and that sum is the definition of the dot product. The dot answers the question "how aligned are these two directions?" — a single number, biggest when parallel ( $α = 0$ ), zero when perpendicular ( $α = 9 0^{\circ}$ ), negative when opposed.

Step 4 — Collect the $i$ -component → a cross product appears

Now sort the blue cells that carry a leftover $i$ tag.

WHAT: the picture isolates the two ring hops $j k$ and $k j$ landing on the $i$ -axis with opposite signs, and highlights the surviving combination $y_{1} z_{2} - z_{1} y_{2}$ .

WHY this bracket matters: $y_{1} z_{2} - z_{1} y_{2}$ is exactly the $x$ -component of the cross product $v_{1} \times v_{2}$ . It came from the asymmetric ring rule ( $j k = + i$ but $k j = - i$ ) — the one place where order flips the sign.

By the same sorting for the $j$ and $k$ tags (cyclic symmetry), all three coefficients assemble into: $vector (pq) = w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}$

$w_{1} v_{2}$ — dial of $p$ stretches the arrow of $q$ .
$w_{2} v_{1}$ — dial of $q$ stretches the arrow of $p$ .
$v_{1} \times v_{2}$ — the sideways twist, the only term that flips when you swap.

Step 5 — Why the cross product is the sole troublemaker

WHAT: two side-by-side right-hand-rule pictures. Left: $v_{1} \times v_{2}$ points up. Right: $v_{2} \times v_{1}$ points down — same magnitude, opposite arrow.

WHY the cross product, and not another sideways rule? We again did not choose it — the ring's antisymmetry ( $ij = + k$ , $j i = - k$ ) produced it. The cross product answers "how much do these two arrows twist across each other, and about which axis?" Its magnitude is $∣ v_{1} ∣∣ v_{2} ∣ sin α$ (biggest when perpendicular, zero when parallel), and its direction is set by the right-hand rule — precisely the ring of Step 1.

Recall Why does

$pq \neq = q p$ in one sentence? Because the cross-product term is the only piece of the Hamilton product that reverses sign when you swap the two quaternions ::: and reversing a physical rotation's order really does give a different orientation.

Step 6 — Degenerate & edge cases (never leave a gap)

We must show every corner, so no reader hits an unshown scenario.

The one-picture summary

WHAT: the whole derivation on one canvas — the $4 \times 4$ grid feeds two buckets; the yellow bucket condenses to $w_{1} w_{2} - v_{1} \cdot v_{2}$ (a dial), the blue bucket to $w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}$ (an arrow). One symmetric dot, one antisymmetric cross.

Recall Feynman retelling — the walkthrough in plain words

We had two little machines, each made of one dial and one arrow. To glue them into one machine we multiplied everything by everything — sixteen tiny pieces. Then we sorted the pieces into two boxes. The number box ended up asking "how much do the two arrows agree?" — that's the dot, and it never cares about order. The arrow box ended up with two stretched copies of the arrows plus a sideways twist — that twist is the cross product, and it's the only thing that notices which machine you used first. Point the arrows the same way and the twist vanishes (spins about one axis always commute); point them across each other and the twist is strongest. That single sideways twist is why turning a cube "right then up" differs from "up then right" — and why quaternions can store real 3D rotation in just four numbers.

Connections

Parent: Hamilton product — the formula this page derives.
Cross Product & Right-Hand Rule — the ring of Step 1 is this rule.
Rotation Matrices — SO(3) — the same non-commutativity, written as matrices.
Axis-Angle & Euler Rodrigues — where the $θ /2$ half-angle enters.
Quaternion Kinematics — $\dot q = \tfrac12 q\,\omega$ — every gyro tick is one Hamilton product.
Multiplicative EKF (MEKF) — chains these products for attitude estimation.
Gimbal Lock & Euler Angles — the failure mode quaternions dodge.

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