Foundations — Vector addition — triangle law, parallelogram law

1.1.8 · D1 · Physics › Measurement, Vectors & Kinematics › Vector addition — triangle law, parallelogram law

Is page par kuch bhi assume nahi kiya gaya. Parent note ek daraavne-se formula aur ek direction rule ke saath khatam hota hai, jo letters aur squiggles se bhara hai. Hamaara poora kaam yahaan yeh hai ki un saare symbols ko ek-ek karke earn karein — arrow-hat, lengths, plus sign, angle, sine/cosine/tangent, components, square root — ek aisi order mein jahan har naya symbol sirf pehle se bane waale symbols par lean kare. Last section tak tum parent ke formula ko plain words mein padh paoge.

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1. Arrow-hat: $\vec{A}$

Figure 1.

Figure 1 mein blue arrow dekho. Uska tail wahan hai jahan se shuru hota hai (mota dot), uska head pointy end hai. Ise poori tarah describe karne ke liye tumhe dono bolna hoga — "kitna lamba" aur "kidhar point karta hai". Ek plain number jaise $7$ yeh nahi kar sakta — $7$ kya? $7$ steps kahan?

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2. Arrows ke liye plus sign: $\vec{A}+\vec{B}$

Toh "$+$" symbol ka matlab change hota hai depending on kya uske dono taraf baitha hai:

• do plain numbers ke beech ($3+4$) iska matlab ordinary counting hai, jo $7$ deta hai;
• do arrows ke beech ($\vec{A}+\vec{B}$) iska matlab "pehle ek journey karo phir doosri" hai, jo diagonal $5$-arrow deta hai, $7$ nahi.

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3. Playing field: $x$ aur $y$ axes

Figure 2.

Ise graph paper ki tarah socho jo chaar directions mein phail raha hai (Figure 2). Plane mein ab har point ka ek address hai $(x,y)$: "$x$ steps across, $y$ steps up." Khaas baat yeh hai ki dono axes par dono taraf chal sakte hain — toh ek component ek negative number ho sakta hai (left ya down point karta hua). Hume jald hi is fact ki zaroorat padegi.

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4. Magnitude: $A$ (same letter, hat nahi)

Socho: agar $\vec{A}$ blue arrow hai, toh $A$ woh number hai jo tum ruler se uske saath rakh ke padhoge. Direction phenki, length rakhi.

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5. Tail-to-tail vs head-to-tail — do arrows rakhne ke do tarike

Figure 3.

Figure 3 ki left picture (head-to-tail): orange arrow $\vec{B}$ wahan se continue hota hai jahan blue arrow khatam hua. Yeh literally hai "journey $\vec{A}$ karo, phir journey $\vec{B}$ karo." Green closing arrow resultant hai.

Figure 3 ki right picture (tail-to-tail): dono arrows same dot se start karte hain, aur hum poora parallelogram complete karte hain — $\vec{B}$ ko $\vec{A}$ ke head tak copy karo, $\vec{A}$ ko $\vec{B}$ ke head tak copy karo (dashed sides). Shared corner se green diagonal resultant hai. Tum do originals ke beech shaded wedge dekh sakte ho; woh wedge hi woh angle hai jo hum aage naam denge.

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6. Beech ka angle: $\theta$

Figure 3 ki right picture phir dekho: common tail par shaded wedge hi $\theta$ hai. Hum ise degrees mein measure karte hain (ek poora chakkar $360^{\circ }$ hai, ek square corner $90^{\circ }$ hai). Kyunki humne $x$-axis ko $\vec{A}$ ke saath align kiya (Section 3), $\theta$ whi bhi angle hai jo $\vec{B}$ $x$-axis ke saath banaata hai.

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7. Right triangle, opposite aur adjacent — andar chhupa hua shape

Figure 4.

Arrows add karne ke liye yeh kyun matter karta hai? Kyunki jab hum tilted arrow ke head se seedha neecha line drop karte hain, toh hum ek right triangle manufacture karte hain (Figure 4). Us triangle ki do legs exactly arrow ke horizontal aur vertical parts hain — components jo hum aage milenge.

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8. Sine, cosine, tangent — angle ko ratios mein convert karna

Figure 4 par visualize karo: length $1$ ki hypotenuse ke liye, woh floor par jo shadow daalti hai woh hi $\cos \theta$ hai, aur uski height hi $\sin \theta$ hai. Arrow ko flat tilt karo ($\theta =0$): shadow poori length hai ($\cos 0=1$), height zero hai ($\sin 0=0$). Seedha khada karo ($\theta =90^{\circ }$): shadow gayab ho jaati hai ($\cos 90^{\circ }=0$), height full hai ($\sin 90^{\circ }=1$). Yahi wajah hai ki parent ke sanity checks kaam karte hain.

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9. Har arrow ke components — aur $R_x,R_y$ build karna

Ab hum Sections 3, 7 aur 8 ka fayda uthate hain. $x$-axis ko $\vec{A}$ ke saath rakh ke:

Figure 5 $\vec{B}$ akela dikhata hai, $\theta$ par tilted, apne do shadows ke saath.

Figure 5.

Components ko Section 8 ki definitions se seedha padho (hypotenuse $=B$):

• $\vec{A}$ $x$-axis ke saath lies karta hai, toh uske shadows $(A,0)$ hain — sab across, upar kuch nahi.
• $\vec{B}$ $\theta$ se tilted hai, toh uska horizontal shadow $B\cos \theta$ (adjacent) hai aur vertical shadow $B\sin \theta$ (opposite) hai: $(B\cos \theta ,B\sin \theta )$.

Ab Section 2 ke operator se arrows add karo. Kyunki horizontal journeys kabhi vertical ko disturb nahi karti, hum across-parts alag aur up-parts alag add karte hain:

R_y=\underbrace{0}_{\vec A\text{ up}}+\underbrace{B\sin\theta}_{\vec B\text{ up}}=B\sin\theta.$$ Toh resultant ke components hain $$\boxed{R_x=A+B\cos\theta,\qquad R_y=B\sin\theta.}$$ Yeh parent ke Steps 1–2 ka poora engine hai — ab sirf quote nahi kiya, balki *derive* kiya gaya. Chhote subscripts ka matlab hai "across part" ($x$) aur "up part" ($y$). Axes ki wajah se (Section 3), har component ek **sign carry** karta hai: - $R_x>0$ right point karta hai, $R_x<0$ left point karta hai, - $R_y>0$ upar point karta hai, $R_y<0$ neecha point karta hai. $(R_x,R_y)$ sign ka pair batata hai ki resultant chaar **quadrants** mein se kis mein hai: | $R_x$ | $R_y$ | quadrant | arrow point karta hai... | |:---:|:---:|:---:|:---| | $+$ | $+$ | I | up-right | | $-$ | $+$ | II | up-left | | $-$ | $-$ | III | down-left | | $+$ | $-$ | IV | down-right | **Figure 6.** ![[deepdives/dd-physics-1.1.08-d1-s06.png]] --- ## 10. Pythagoras: components ko wapas ek length mein jodna > [!formula] Pythagoras' theorem > Ek right triangle mein, $\;\text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2$. Components ke liye: > $$R=\sqrt{R_x^2+R_y^2}.$$ > [!intuition] Humein iska zaroorat kyun hai > Resultant ko sideways part $R_x$ aur up part $R_y$ mein split karne ke baad, woh do parts ek right triangle ki do legs banate hain — aur resultant khud hypotenuse hai. Pythagoras woh tool hai jo "$R_x$ across, $R_y$ up" ko wapas ek single length $R$ mein convert karta hai. Chhota "$\sqrt{\phantom{x}}$" **square root** hai — yeh squaring ko undo karta hai. Hum har leg ko square karte hain, add karte hain, phir length wapas paane ke liye square root lete hain. Notice karo ki **squaring signs erase kar deta hai** ($(-4)^2=16$), toh $R$ positive nikalta hai chahe arrow kisi bhi quadrant mein ho — exactly jaise ek length ko hona chahiye. --- ## 11. Length formula ko sath mein rakhna (algebra poori tarah dikhaya gaya) Section 9 ke boxed components ko Pythagoras mein substitute karo: $$R=\sqrt{(A+B\cos\theta)^2+(B\sin\theta)^2}.$$ **Pehla bracket expand karo** — $(p+q)^2=p^2+2pq+q^2$ use karke $p=A,\ q=B\cos\theta$ ke saath: $$(A+B\cos\theta)^2 = A^2+2AB\cos\theta+B^2\cos^2\theta.$$ **Doosra bracket expand karo:** $$(B\sin\theta)^2 = B^2\sin^2\theta.$$ **Dono add karo:** $$R^2 = A^2+2AB\cos\theta+B^2\cos^2\theta+B^2\sin^2\theta.$$ **Do $B^2$ pieces group karo** aur $B^2$ factor out karo: $$R^2 = A^2+2AB\cos\theta+B^2\big(\cos^2\theta+\sin^2\theta\big).$$ Ab woh ek identity use karo jo har right triangle obey karta hai, $\cos^2\theta+\sin^2\theta=1$ (yeh sirf length $1$ ki hypotenuse par Pythagoras hai): $$R^2 = A^2+B^2+2AB\cos\theta.$$ > [!formula] Resultant ka magnitude > $$\boxed{R=\sqrt{A^2+B^2+2AB\cos\theta}}$$ Ab tum *dekh* sakte ho ki famous $2AB\cos\theta$ term kahan se paida hota hai: yeh cross-term $2\cdot A\cdot B\cos\theta$ hai bracket $(A+B\cos\theta)$ ko square karne se. Yeh measure karta hai ki $\vec{B}$ kitna $\vec{A}$ ke *saath* point karta hai — bada aur positive jab align hों, negative jab oppose karein. --- ## 12. Direction angle $\alpha$ — aur sahi quadrant choose karna > [!definition] Alpha > $\alpha$ (Greek letter "alpha") woh angle hai jo **resultant** $\vec{R}$ **$x$-axis** ke saath banaata hai. Kyunki humne deliberately $x$-axis ko $\vec{A}$ ke saath lay kiya (Section 3), yeh *bilkul wahi* hai jaise "$\vec{R}$ ka $\vec{A}$ ke saath angle" — jaise parent page mein phrase kiya gaya hai. Alignment hi woh cheez hai jo do descriptions ko identical banaati hai. Hum ise Section 8 ke steepness tool se dhundhte hain: $$\tan\alpha=\frac{R_y}{R_x}\quad(\text{vertical part over horizontal part}).$$ > [!mistake] "$\tan\alpha=R_y/R_x$ akela direction deta hai." > **Kyun sahi lagta hai:** yeh slope hai, aur slope ka matlab direction hai. **Flaw:** $\tan$ ek arrow aur uske exact opposite ke liye *same* number deta hai (jaise up-right vs down-left) — toh bare ratio dono mein fark nahi kar sakta. **Fix:** pehle $R_x,R_y$ ke **signs** padho quadrant fix karne ke liye (Section 9 table), *phir* tilt par trust karo. Agar $R_x<0$ hai, toh calculator jo return kare usme $180^\circ$ add karo. > [!mistake] "Formula tab bhi kaam karta hai jab $R_x=0$ ho." > **Kyun sahi lagta hai:** hum bas numbers plug in karte hain. **Flaw:** agar $R_x=0$ hai toh resultant **seedha upar ya seedha neecha** hai, aur $R_y/R_x$ zero se divide karta hai — $\tan\alpha$ *undefined* hai, koi angle nahi nikalta. **Fix:** is vertical case ko haath se handle karo: agar $R_x=0$ aur $R_y>0$ toh $\alpha=+90^\circ$ (seedha upar); agar $R_x=0$ aur $R_y<0$ toh $\alpha=-90^\circ$ (seedha neecha); agar dono $0$ hain toh resultant zero arrow hai aur uski koi direction nahi. **Figure 7.** ![[deepdives/dd-physics-1.1.08-d1-s07.png]] > [!intuition] Yeh yahaan kyun matter karta hai > Parent page par $\vec{B}$ ek friendly angle par baitha hai toh $R_x=A+B\cos\theta$ positive hai aur sab kuch quadrant I mein land karta hai — koi correction needed nahi. Lekin jis pal tum aisi arrows add karo jo left ya down point karti hain (koi bhi real physics problem), tum *zaroor* sign rule aur $R_x=0$ case ko fold in karo, warna tumhara khoobsoorat $R$ galat taraf point karega. --- ## 13. Boxed formula padhna — har symbol earn kiya gaya Ab parent ka headline result plain words mein padha jaata hai: $$R=\sqrt{\underbrace{A^2}_{(\text{length of }\vec A)^2}+\underbrace{B^2}_{(\text{length of }\vec B)^2}+\underbrace{2AB\cos\theta}_{\text{overlap term}}}$$ - $R$ — resultant arrow ki **length** (Section 10), - $A,B$ — do arrows ki lengths (Section 4), - $\cos\theta$ — kitna $\vec B$ $\vec A$ ke *saath* point karta hai (Section 8), $\theta$ tail-to-tail angle ke saath (Section 6), - $2AB\cos\theta$ cross-term — $(A+B\cos\theta)^2$ expand karne se paida hota hai (Section 11), - square aur square-root — Pythagoras (Section 10). Aur uska saathi, direction: $$\tan\alpha=\frac{R_y}{R_x},\qquad \alpha=\vec R\text{ ka tilt }\vec A\text{ se measure kiya gaya}\ (\text{Section 12, signs aur }R_x=0\text{ case dhyan mein rakho}).$$ $2AB\cos\theta$ term woh "kya arrows ek doosre ki madad karte hain ya ladte hain?" term hai: jab woh align hote hain ($\cos\theta=1$) yeh sabse zyaada add karta hai; jab woh oppose karte hain ($\cos\theta=-1$) yeh subtract karta hai. Ab tumhare paas har piece hai. --- ## Prerequisite map ```mermaid graph TD HAT["Arrow-hat vector A"] --> PLUS["Plus operator for arrows"] HAT --> MAG["Magnitude A length only"] AXES["x and y axes with signs"] --> ALIGN["Align x-axis along A"] ALIGN --> COMP["Components Rx and Ry"] HAT --> PLACE["Head-to-tail and tail-to-tail"] PLACE --> THETA["Angle between theta"] THETA --> TRIG["Sine cosine tangent"] MAG --> COMP TRIG --> COMP PLUS --> COMP COMP --> PYTH["Pythagoras gives length"] COMP --> SIGN["Quadrant sign rule"] PYTH --> RES["Resultant magnitude R"] TRIG --> DIR["Direction angle alpha"] SIGN --> DIR RES --> TOPIC["Triangle and parallelogram laws"] DIR --> TOPIC ``` Dekho kaise components crossroads par baithe hain: unhe magnitude idea chahiye, trig ratios chahiye, "$+$" operator *aur* signed, $\vec{A}$-aligned axes chahiye, aur woh *dono* answer ki length aur direction ko feed karte hain. Yahi wajah hai ki [[Resolution of a vector into components]] aur [[Vectors — components and unit vectors]] natural agli stops hain. --- ## Equipment checklist $\vec{A}$ mein hat tumhe kya batata hai? ::: Yeh ek vector hai — ek arrow jisme ek length aur ek direction dono hain. Plain letter $A$ (bina hat ke) ka matlab kya hai? ::: Sirf $\vec{A}$ ki length (magnitude) — ek non-negative number. Do arrows $\vec{A}+\vec{B}$ ke beech "$+$" ka matlab kya hai? ::: Ek naya operation: $\vec{B}$ ka tail $\vec{A}$ ke head par rakho aur start se finish tak arrow draw karo — yeh direction rakhta hai, toh yeh "lengths add karo" nahi hai. Hum $x$-axis ko $\vec{A}$ ke saath kyun lagate hain? ::: Yeh ek free choice hai jo $\vec{A}$ ko horizontal banati hai components $(A,0)$ ke saath, toh "$\vec{A}$ ke saath angle" equals "$x$-axis ke saath angle." $x$ aur $y$ axes tumhe kya dete hain? ::: "Across" vs "up" aur kaunsi side positive hai ka ek shared agreement — taaki components ko signed numbers milein. Triangle law ke liye do arrows kaise place karte ho? ::: Head-to-tail: doosre ka tail pehle ke head par. Parallelogram law ke liye unhe kaise place karte ho? ::: Ek common point se tail-to-tail; parallelogram complete karo; resultant diagonal hai. Har formula mein $\theta$ exactly kya hai? ::: Do vectors ke beech ka angle, tail-to-tail measure kiya gaya. $\cos\theta$ right triangle mein kaun sa ratio equals karta hai? ::: adjacent ÷ hypotenuse. $\sin\theta$ kaun sa ratio equals karta hai? ::: opposite ÷ hypotenuse. $\tan\theta$ kaun sa ratio equals karta hai, aur yeh kya measure karta hai? ::: opposite ÷ adjacent — steepness (slope), direction ke liye use hota hai. Jab $x$-axis $\vec{A}$ ke saath ho toh $\vec{A}$ aur $\vec{B}$ ke components kya hain? ::: $\vec{A}=(A,0)$ aur $\vec{B}=(B\cos\theta,\,B\sin\theta)$. Un components se $R_x$ aur $R_y$ kaise aate hain? ::: Across-parts aur up-parts alag-alag add karo: $R_x=A+B\cos\theta$, $R_y=B\sin\theta$. $2AB\cos\theta$ term kahan se aata hai? ::: Cross-term jab tum Pythagoras ke andar $(A+B\cos\theta)^2$ expand karte ho. $R_x$ aur $R_y$ ko ek length mein re-assemble karne ka tool kaun sa hai? ::: Pythagoras: $R=\sqrt{R_x^2+R_y^2}$. $\alpha$ kya hai, aur $\tan\alpha=R_y/R_x$ akele kyun kaafi nahi hai? ::: Resultant ki direction $\vec{A}$ ($x$-axis) se measure ki gayi; tan ek arrow aur uske opposite mein fark nahi kar sakta, toh signs se quadrant fix karo ($180^\circ$ add karo agar $R_x<0$). Jab $R_x=0$ ho toh kya karte ho? ::: $R_y/R_x$ undefined hai; $\alpha=+90^\circ$ set karo agar $R_y>0$, $\alpha=-90^\circ$ agar $R_y<0$, aur koi direction nahi agar dono $0$ hain. $R=\sqrt{A^2+B^2+2AB\cos\theta}$ mein $2AB\cos\theta$ term kya kar raha hai? ::: Yeh measure kar raha hai ki do arrows kitna ek doosre ki help (align) ya fight (oppose) karte hain. ## Connections - [[Vector addition — triangle law, parallelogram law]] (parent — yeh page uska ground floor hai) - [[Resolution of a vector into components]] (Sections 7–9 ek poori method mein) - [[Vectors — components and unit vectors]] ($\hat i,\hat j$ language for components) - [[Law of cosines]] (magnitude formula triangle-geometry clothing mein) - [[Scalar (dot) product]] ($\cos\theta$ overlap term ke liye ek compact machine)