Foundations — Cross product — formula, direction (right-hand rule), torque - area calculation

1.1.12 · D1 · Physics › Measurement, Vectors & Kinematics › Cross product — formula, direction (right-hand rule), torque

Parent note mein jo bhi hai, woh sab chhoti-chhoti ideas ke ek pile se bana hai. Yeh page us pile ko bilkul scratch se kholta hai. Hum order mein jaate hain, isliye jo bhi symbol tum miloge, woh usse pehle wali line mein already earn ho chuka hoga.

0. Sabse raw idea: ek vector

Ek plain number — jaise temperature $20$ — sirf size rakhta hai. Ek vector mein size aur ek heading dono hoti hain, jaise "$20$ metres, north-east aur thoda upar".

IS TOPIC KO ISKA KYU CHAHIYE: cross product do vectors khaata hai aur teesra banata hai. Agar tumhe yeh poori tarah se comfortable nahi hai ki ek arrow ek direction carry karta hai, toh iske baad kuch bhi sense nahi banega.

Arrows ki puri kahani ke liye dekho Vectors — addition, components, unit vectors।

1. Vector ki length: $A$ (bina hat ke)

Toh $\vec{A}$ arrow hai; $A$ bas itna hai ki woh arrow kitna lamba hai. Parent note mein magnitude formula $|\vec{A}\times \vec{B}|=AB\sin \theta$ do lengths $A$ aur $B$ ko multiply karta hai — woh yahi hat-free numbers hain.

2. Teen axes aur unit vectors $\hat{i},\hat{j},\hat{k}$

Arrows ke saath arithmetic karne ke liye pehle hum teen reference directions pe agree karte hain.

IS TOPIC KO ISKA KYU CHAHIYE: parent ka poora component machine $\hat{i},\hat{j},\hat{k}$ ke terms mein likha gaya hai, aur famous cycle $\hat{i}\times \hat{j}=\hat{k}$ tab tak sense nahi banta jab tak tum in teen pointers ko picture nahi kar sakte.

3. Components: $A_x,A_y,A_z$

Jab axes ho jaayein, kisi bhi arrow ko describe kiya ja sakta hai ki woh har axis ke along kitna reach karta hai.

Har component ek plain (possibly negative) number hai: negative ka matlab bas "us axis ke doosri taraf chalo".

IS TOPIC KO ISKA KYU CHAHIYE: parent mein §4 ka determinant formula bas in chhe numbers $A_x,A_y,A_z,B_x,B_y,B_z$ ki bookkeeping hai, kuch nahi.

4. Tail-to-tail aur angle $\theta$

IS TOPIC KO ISKA KYU CHAHIYE: magnitude $AB\sin \theta$ poori tarah is angle par depend karta hai. Zyada spread → bada cross product; koi spread nahi → zero.

5. Trig switch $\sin \theta$ — aur kyun $\cos \theta$ nahi

Yahan story mein ek tool enter ho raha hai, isliye hum exactly kyun yeh tool aur koi doosra nahi yeh bataate hain.

Pooch: "$\vec{B}$ ka kitna hissa $\vec{A}$ ke across point karta hai, uske along nahi?" Jawaab hai $B\sin \theta$. Wahi across-ness precisely woh "spread" hai jo cross product measure karta hai.

Alignment cousin $\cos \theta$ ko Dot product — formula, projection, work calculation mein develop kiya gaya hai.

6. Perpendicular aur right-hand rule

Cross product ki direction dono input arrows ke perpendicular hoti hai ek saath. Teen dimensions mein aisi do directions hoti hain (seedha page ke bahar, ya seedha andar). Right-hand rule decide karta hai kaun si.

IS TOPIC KO ISKA KYU CHAHIYE: ek ghoomta darwaza, ek spinning wheel aur ek facing surface sab ek axis direction chahte hain. Ek bare number "is taraf" nahi bol sakta. Perpendicular arrow bol sakta hai — yahi poori wajah hai ki cross product ek vector return karta hai. Yahi handedness Angular momentum — L = r × p aur Magnetic force — F = qv × B mein dobara aati hai.

7. Cross symbol $\times$ aur vector-arrow output

Compare karo: plain numbers ke saath $A\times B$ bas $A$ times $B$ hai. Arrows ke saath $\vec{A}\times \vec{B}$ ek bilkul naya arrow hai. Hats batate hain ki tum kis world mein ho.

8. Parallelogram aur triangle ka area

KYU: parent ka Example 2 (triangle area) exactly yahi fact action mein hai.

9. Determinant grid

Tumhe ise use karne ke liye determinants master nahi karne padte — parent note pattern walk karta hai — lekin poori mechanics Determinants — 3×3 expansion mein rehti hain. Yahan se carry karne wala ek rule: middle ($\hat{j}$) term hamesha minus sign leta hai.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.