Foundations — Resolution of vectors — into components (any axes)

1.1.9 · D1 · Physics › Measurement, Vectors & Kinematics › Resolution of vectors — into components (any axes)

Kisi bhi vector ko resolve karne se pehle, parent note ek poora toolbox quietly use karta hai: arrows, angles, right triangles, sine, cosine, unit vectors, dot products. Yeh page un sabko zero se build karta hai, us order mein jisme ek doosre par depend karte hain. Agar parent note mein koi symbol aata hai, toh usse pehle yahan define kiya gaya hai.

1. Arrow: vector hota kya hai

Socho ek toy car ko push karna. Sirf "Maine 5 units force se push kiya" kehna kaafi nahi — kis direction mein push kiya? Ek vector dono ka ek saath jawab deta hai.

• Hum vector ko $\vec{A}$ likhte hain — upar chhoti arrow ka matlab hai "yeh ek vector hai, sirf plain number nahi."
• Ek plain number (jaise temperature, ya sirf length) bina direction ke scalar kehlata hai.

2. Magnitude: arrow ki length

Arrow ko graph paper par rakho. Ruler lo. Jo number aata hai woh $A$ hai. Yeh kabhi negative nahi hota — length zero se kum nahi ho sakti.

Topic ko iska kyun zaroorat hai: parent ke key formulas $A_x=A\cos \theta$ aur $A_y=A\sin \theta$ is length $A$ ko ek fraction se multiply karte hain. Agar scale karne ke liye koi length hi na ho, toh resolve karne ko kuch hoga hi nahi.

3. Angle $\theta$: arrow kis taraf jhukta hai

• $\theta =0^{\circ }$ matlab arrow seedha right ki taraf point kar raha hai.
• $\theta =90^{\circ }$ matlab seedha upar.
• Chhota ${}^{\circ }$ symbol matlab "degrees" — ek poora chakkar $360^{\circ }$ hota hai.

Topic ko iska kyun zaroorat hai: $\theta$ woh akela number hai jo batata hai ki arrow kaise tilted hai, aur isliye kitna "leak" karta hai sideways ki taraf aur kitna upar ki taraf.

4. Right triangle: har vector ke andar chhupa hua secret shape

Teeno sides ke naam angle $\theta$ ke relative hain:

• Hypotenuse — sabse lambi side, hamesha square corner ke opposite. Yahan yeh arrow khud hai, length $A$.
• Adjacent — woh side jo angle $\theta$ ke saath (touch karke) hai, x-axis ke along flat padi hai. Yeh $A_x$ ban jaayegi.
• Opposite — woh side jo $\theta$ ke samne hai, vertically khadi hai. Yeh $A_y$ ban jaayegi.

Topic ko iska kyun zaroorat hai: resolution hai hi is triangle ki do chhoti sides padhna. Neeche ke har formula in teeno sides ka ek ratio hai.

5. Sine aur cosine: rules jo angle ko ratio mein convert karte hain

Ab hume ek aisa tool chahiye jo pooche: "diya hua tilt $\theta$, kitna fraction arrow ka flat pada hai, aur kitna fraction khada hai?" Sine aur cosine exactly wahi do fractions hain.

Definitions ko rearrange karne se seedha parent ka core result milta hai. Kyunki $\cos \theta =A_x/A$, dono sides ko $A$ se multiply karo: $A_x=A\cos \theta ,A_y=A\sin \theta .$

6. Components $A_x$ aur $A_y$: do chhipi hui arrows

• Subscript $x$ in $A_x$ sirf direction label karta hai — "A ka woh hissa jo x ke along hai."
• Ek component negative ho sakta hai: agar arrow left ki taraf point kare, toh $A_x$ negative hoga, yaani "minus-x direction mein pahunch raha hai."

Topic ko iska kyun zaroorat hai: yeh do numbers resolution ka poora point hain. Inke aa jaane ke baad, vectors ko numbers ko add karke add kar sakte ho.

7. Pythagoras: arrow ko uske pieces se wapas banana

Agar hum do components se shuru karke original length wapas chahte hain, toh hum woh rule use karte hain jo right triangle ki teeno sides ko aapas mein connect karta hai.

Chhota $2$ exponent matlab "khud se multiply karo" ($A_x^2=A_x\times A_x$).  ​ symbol squaring ko undo karta hai — yeh poochta hai "kaun sa number, squared hone par yeh dega?"

Topic ko iska kyun zaroorat hai: yeh resolution ki inverse direction hai — do numbers se length-and-angle ki taraf jaana. Parent ise "recovering magnitude" kehta hai.

8. Tangent aur arctangent: angle wapas recover karna

Components se $\theta$ wapas paane ke liye hume ek aisa tool chahiye jo do chhoti sides ko ek doosre se relate kare.

Phir $\theta$ khud nikalane ke liye hum ulta sawaal poochte hain, "kaun sa angle is tangent wala hai?" — woh reverse operation ${\tan }^{-1}$ likha jaata hai (padho "arctangent" ya "inverse tan"): $\theta ={\tan }^{-1}\left(\frac{A_y}{A_x}\right).$

Yahan chhota ${}^{-1}$ ka matlab "one over" nahi — matlab hai "undo karo." ${\tan }^{-1}$ ek ratio leta hai aur woh angle deta hai jisne use produce kiya.

9. Unit vectors $\hat{x}$, $\hat{y}$, $\hat{n}$: exactly ek ki length wale arrows

Unit vector ko ek signpost socho: uske paas koi size nahi, sirf direction hai. Ise kisi number se multiply karo toh kisi bhi length par scale ho jaata hai.

Topic ko iska kyun zaroorat hai: $\hat{n}$ parent ko x–y axes se kisi bhi axis tak generalize karne deta hai. Kisi tirchi direction ke along shadow nikaalने के लिए, pehle us direction ko unit vector se name karo.

Poori treatment ke liye dekho Unit vectors and Cartesian coordinates.

10. Dot product: ek arrow ka shadow doosre par measure karna

Parent ka aakhri tool yeh poochta hai: "$\vec{A}$ ka kitna hissa direction $\hat{n}$ ki taraf point karta hai?"

Dot ($\cdot$) is operation ka symbol hai; yeh hamesha do vectors leta hai aur ek plain number (scalar) deta hai. Notice karo yeh cosine reuse karta hai — kyunki shadow phir se "adjacent-over-hypotenuse" hai, bas x-axis ki jagah ek tilted direction ke along measure kiya gaya.

Topic ko iska kyun zaroorat hai: yeh kisi bhi axis par resolution hai, sirf horizontal/vertical nahi. Poori details Dot product & scalar projection mein hain.

Prerequisite map

Baayein har box solid hona chahiye pehle right waale topic ko samajhne se pehle. Arrows follow karo aur tumne poora toolbox zero se build kar liya.

Equipment checklist

Connections

• Parent: Resolution of vectors — into components (any axes)
• Unit vectors and Cartesian coordinates
• Dot product & scalar projection
• Vectors — addition (parallelogram & triangle law)