Quaternion product — Hamilton product: Foundations

3.5.7 · D1 · Physics › Guidance, Navigation & Control (GNC) › Quaternion product — Hamilton product

Is page par koi bhi assumption nahi hai. Isse pehle ki tum "do quaternions multiply karo" padh sako, tumhe har woh symbol khud se samajhna hoga jo aata hai: scalar kya hota hai, vector kya hota hai, axis aur angle kya hote hain, dot product aur cross product kya measure karte hain, aur woh strange letters $i, j, k$ actually kya karte hain. Hum inhe ek-ek karke, ek ke upar doosra, build karte hain.

0. Numbers jin par tumhara pehle se bharosa hai — scalar

Yeh kyun chahiye: ek quaternion mein ek scalar slot hota hai (jise $w$ kehte hain) jo store karta hai "rotation mein koi turn nahi kitna hai." Jab do rotations similar directions mein point karte hain, scalar part badhta hai. Yeh woh anchor hai jis par baaki sab tikaa hota hai.

1. Direction wala ek arrow — vector

Figure — Quaternion product — Hamilton product

Picture mein kya dikhta hai: teen mutually perpendicular chalk axes ( $x$ daayein, $y$ page ke andar, $z$ upar), aur ek blue arrow $v$ jiska shadow har axis par uske teen numbers deta hai. Woh shadows components $x, y, z$ hain.

Yeh topic ko kyun chahiye: ek quaternion mein teen vector slots $(x, y, z)$ hote hain jo milkar woh axis store karte hain jiske around rotation twist karta hai. Scalar $w$ aur vector $v$ milkar poora quaternion banta hai: $q = (w, v)$ .

2. Axis aur angle — ek turn kaise describe karte hain

Picture mein kya dikhta hai: ek yellow axis arrow $\overset{n}{^}$ aur ek pink curved arrow jo turn angle $θ$ ko iske around sweep karta hai. Apna right thumb $\overset{n}{^}$ ki taraf point karo; tumhari curling fingers positive turn direction dikhati hain.

Yeh topic ko kyun chahiye: parent ka Example 2 seedha axis aur angle se quaternion banata hai $q = (cos \frac{θ}{2}, sin \frac{θ}{2} \overset{n}{^})$ ke zariye. Jab tak axis aur angle solid nahi hain, tum woh line nahi padh sakte. (Half-angle kahan se aata hai yeh Axis-Angle & Euler Rodrigues ka kaam hai.)

3. Dot product — "do arrows kitna agree karte hain?"

Yahi tool kyun, koi aur kyun nahi? Dot product exactly ek sawaal ka jawaab deta hai: "yeh do arrows kitne aligned hain?" Yeh sabse bada hota hai jab woh same direction mein point karte hain ( $cos 0 = 1$ ), zero hota hai jab woh perpendicular hote hain ( $cos 9 0^{\circ} = 0$ ), aur negative hota hai jab woh oppose karte hain. Perpendicular-gives-zero wala fact yahi wajah hai ki parent ke Worked Example 1 mein $v_{1} \cdot v_{2} = 0$ hai $i$ aur $j$ ke liye.

Yeh topic ko kyun chahiye: har Hamilton product ka scalar part hota hai $w_{1} w_{2} - v_{1} \cdot v_{2}$ . Woh minus-dot axis alignment ka direct measurement hai.

4. Cross product — "twist jo sideways point karta hai"

Picture mein kya dikhta hai: do chalk arrows $a$ (blue) aur $b$ (pink) flat pade hain, aur unka cross product $a \times b$ (yellow) board se seedha upar khada hai. Right-hand fingers ko $a$ se $b$ ki taraf point karo; thumb $a \times b$ hai.

Yahi tool kyun, koi aur kyun nahi? Humein kuch aisa chahiye jo "twist across" capture kare — ek turning tendency jiska apna direction ho. Sirf cross product 3D mein yeh karta hai. Yahi wajah hai ki Hamilton product ke vector part mein $v_{1} \times v_{2}$ term hota hai.

5. Imaginary units $i, j, k$

6. Norm, conjugate, inverse — quaternion ko "valid" rakhna

Yeh topic ko kyun chahiye: ek vector ko rotate karne ke liye "sandwich" $q u q^{- 1}$ use hota hai — jab tak tum nahi jaante ki $q^{- 1}$ aur $\overset{q}{ˉ}$ kya matlab rakhte hain, tum yeh nahi padh sakte. Aur parent page par "always renormalize" wali galti tab hi samajh aati hai jab tum $∣ q ∣$ jaante ho.

Prerequisite map

Ise upar se neeche padho: plain numbers aur arrows quaternion banate hain; dot aur cross product ke do halves banate hain; units $i, j, k$ cross-product ka right-hand rule algebra mein laate hain; aur norm/conjugate/inverse tumhe quaternion use karne dete hain rotate karne ke liye.

Equipment checklist

Daayein taraf cover karo aur khud ko test karo.

Ek scalar sirf yeh carry karta hai aur bas yahi

size (magnitude), koi direction nahi

Ek vector ko is tarah draw kiya jaata hai

ek length aur direction wala arrow,

(x, y, z)

likhte hain

Ek unit vector (hat, jaise

\overset{n}{^}

) ki length hoti hai

exactly

1

Ek turn describe karne ke liye konsi do cheezein chahiyen

ek axis

\overset{n}{^}

aur ek angle

θ

a \cdot b

zero hota hai jab arrows hote hain

perpendicular (

9 0^{\circ}

par)

Dot product order ki parwah karta hai?

Nahi — yeh symmetric hai,

a \cdot b = b \cdot a

a \times b

point karta hai

dono ke perpendicular, right-hand rule se

Cross product order ki parwah karta hai?

Haan —

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

Kaun sa ingredient Hamilton product ko non-commutative banata hai

sirf cross product term

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

equals

- 1

ij

equals

k

; aur

j i

equals

- k

Ek unit quaternion ka norm hota hai

∣ q ∣ = 1

(w, v)

ka conjugate

\overset{q}{ˉ}

hota hai

(w, - v)

Ek unit quaternion ke liye

q^{- 1}

equals

\overset{q}{ˉ}

Connections

Parent: Hamilton product — jahan yeh saare symbols milkar multiply hote hain.
Cross Product & Right-Hand Rule — $ij = k$ aur non-commutativity ki geometric root.
Axis-Angle & Euler Rodrigues — jahan axis, angle aur half-angle quaternion mein enter karte hain.
Rotation Matrices — SO(3) — same rotations $3 \times 3$ matrices ke roop mein likhe gaye.
Quaternion Kinematics — $\dot q = \tfrac12 q\,\omega$ — norm aur product ko har gyro tick par use karta hai.
Multiplicative EKF (MEKF) — attitude filters jo in products ko chain karte hain.
Gimbal Lock & Euler Angles — woh problem jisse bachne ke liye quaternions banaye gaye the.