Quaternion product — Hamilton product — Hinglish notes

3.5.7 · Physics › Guidance, Navigation & Control (GNC)

Quaternion kya hota hai

$ij k = - 1$ aur squaring rules se, pairwise products aate hain (yahi poora engine hai):

$ij = k, j k = i, k i = j (cyclic, right-handed)$ $j i = - k, k j = - i, ik = - j (anti-cyclic)$

Hamilton product ko scratch se derive karna

Formula Derivation — bas distribute karo

Maano $p = w_{1} + x_{1} i + y_{1} j + z_{1} k$ aur $q = w_{2} + x_{2} i + y_{2} j + z_{2} k$ . Term-by-term multiply karo (4 terms par FOIL → 16 products) aur multiplication table substitute karo:

$pq = w_{1} w_{2} + w_{1} (x_{2} i + y_{2} j + z_{2} k) + (x_{1} i + y_{1} j + z_{1} k) w_{2} + (9 cross terms)$

Scalar part collect karo. Sirf unit-squared terms se scalars aate hain: $x_{1} x_{2} i^{2} + y_{1} y_{2} j^{2} + z_{1} z_{2} k^{2} = - (x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2})$ plus $w_{1} w_{2}$ . To $scalar (pq) = w_{1} w_{2} - v_{1} \cdot v_{2}$

$i$ component collect karo. Contributions: $w_{1} x_{2} + x_{1} w_{2}$ (scalar × vector), plus cyclic products jo $i$ par land karte hain: $y_{1} z_{2} (j k) = y_{1} z_{2} i$ aur $z_{1} y_{2} (k j) = - z_{1} y_{2} i$ . To $i$ -coefficient hai $w_{1} x_{2} + x_{1} w_{2} + (y_{1} z_{2} - z_{1} y_{2})$ Last bracket bilkul $v_{1} \times v_{2}$ ka $x$ -component hai. $j, k$ ke liye symmetry se: $vector (pq) = w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}$

Figure — Quaternion product — Hamilton product

Worked example 1 — pure imaginary units

Vector formula use karke $i j$ compute karo.

$p = i$ : $w_{1} = 0, v_{1} = (1, 0, 0)$ . Kyun? $i$ ek pure-vector unit quaternion hai.
$q = j$ : $w_{2} = 0, v_{2} = (0, 1, 0)$ .
scalar $= 0 - (1, 0, 0) \cdot (0, 1, 0) = 0$ . Kyun? $v_{1} ⊥ v_{2}$ hai, dot 0 hai.
vector $= 0 + 0 + (1, 0, 0) \times (0, 1, 0) = (0, 0, 1)$ . Kyun? right-hand cross se $\overset{z}{^}$ milta hai.
Result $= (0, (0, 0, 1)) = k$ . ✅ $ij = k$ se match karta hai.

Ab $j i$ : cross flip hokar $(0, 0, - 1)$ ho jaata hai → result $- k$ . Yeh step kyun? Cross product anti-commutative hota hai, isliye order badalne par sign palat jaata hai. Yeh prove karta hai ki $ij = - j i$ .

Worked example 2 — do real rotations combine karna

$z$ ke around $9 0^{\circ}$ rotate karo, phir $x$ ke around $9 0^{\circ}$ . Axis $\overset{n}{^}$ ke around angle $θ$ ke liye unit quaternion hai $q = (cos \frac{θ}{2}, sin \frac{θ}{2} \overset{n}{^})$ .

$q_{z} = (cos 4 5^{\circ}, (0, 0, sin 4 5^{\circ})) = (\frac{1}{2}, 0, 0, \frac{1}{2})$
$q_{x} = (\frac{1}{2}, \frac{1}{2}, 0, 0)$

Combined rotation (" $z$ ke baad $x$ ") = $q_{x} q_{z}$ (sabse left wala last lagta hai). Kyun? Hamilton convention: $q_{total} = q_{last} q_{first}$ .

$w = w_{1} w_{2} - v_{1} \cdot v_{2} = \frac{1}{2} - (\frac{1}{2}, 0, 0) \cdot (0, 0, \frac{1}{2}) = \frac{1}{2} - 0 = \frac{1}{2}$
$v = w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}$ $= \frac{1}{2} (0, 0, \frac{1}{2}) + \frac{1}{2} (\frac{1}{2}, 0, 0) + (\frac{1}{2}, 0, 0) \times (0, 0, \frac{1}{2})$ $= (0, 0, \frac{1}{2}) + (\frac{1}{2}, 0, 0) + (0, - \frac{1}{2}, 0) = (\frac{1}{2}, - \frac{1}{2}, \frac{1}{2})$

To $q_{x} q_{z} = (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2}, \frac{1}{2})$ . Check: norm $= 4 \cdot \frac{1}{4} = 1$ ✅ (unit quaternions ka product unit hota hai — isliye gyro integration orientations ko valid rakhta hai).

Vector ko rotate karna

Sandwich kyun, sirf $q u$ kyun nahi? Ek single product scalar aur vector parts ko mix kar deta aur length badal deta. Conjugation $q (\cdot) q^{- 1}$ scalar leakage cancel karta hai aur rotation ko twice-by-half apply karta hai, isliye $q$ mein half-angle $θ /2$ aata hai.

Common mistakes

Recall

Recall Flashables (chhupaao aur khud test karo)

Hamilton product ke do parts kya hain? scalar $= w_{1} w_{2} - v_{1} \cdot v_{2}$ ; vector $= w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}$ .
Non-commutativity ka akela source kya hai? cross product term.
$ij = ?$ $= k$ . $j i = ?$ $= - k$ .
Vector ko rotate kaise karte hain? Unit $q$ ke liye $q u \overset{q}{ˉ}$ .

Recall Feynman: 12-saal ke bachche ko samjhao

Socho do magic "turn cards" hain. Har card, jab tap karo, ek toy ko space mein spin karta hai. Agar tumhe ek aisa card chahiye jo dono spins ek saath kare, to tum do cards ko ek special recipe se jodte ho. Recipe mein do cheezein hain: ek part check karta hai ki dono spins kitna same direction mein point karte hain (woh dot part hai, ek plain number), aur doosra part check karta hai kitna woh ek doosre ke across twist karte hain (woh cross part hai, jo right-hand rule se sideways point karta hai). Kyunki across-twisting is baat par depend karta hai ki aap pehle kaun sa card tap karte ho, card A phir B tapna same nahi hai jaise B phir A — bilkul jaise Rubik's cube ko ghoomana.

Connections

Rotation Matrices — SO(3) — Hamilton product ≡ rotations ki matrix multiplication.
Axis-Angle & Euler Rodrigues — $q$ ke andar half-angle $θ /2$ kahaan se aata hai.
Quaternion Kinematics — $\dot q = \tfrac12 q\,\omega$ — gyro integration isi product ka use karta hai.
Multiplicative EKF (MEKF) — attitude filters Hamilton products chain karte hain.
Gimbal Lock & Euler Angles — woh problem jo quaternions solve karte hain.
Cross Product & Right-Hand Rule — $ij = k$ ki geometric root.

Hamilton product scalar part

w_{1} w_{2} - v_{1} \cdot v_{2}

Hamilton product vector part

w_{1} v_{2} + w_{2} v_{1} + v_{1} \times v_{2}

Which term causes non-commutativity

cross product

v_{1} \times v_{2}

Value of

ij

k

Value of

j i

- k

Fundamental relation defining quaternions

i^{2} = j^{2} = k^{2} = ij k = - 1

Order convention: rotation A then B combined quaternion

q_{B} q_{A}

(right-to-left)

How to rotate vector u by unit quaternion q

u^{'} = q u \overset{q}{ˉ}

Inverse of a unit quaternion

iska conjugate

\overset{q}{ˉ} = (w, - v)

Is the product of two unit quaternions a unit quaternion

Haan,

∣ pq ∣ = ∣ p ∣∣ q ∣