Visual walkthrough — Scalars vs vectors — definition, examples

1.1.6 · D2 · Physics › Measurement, Vectors & Kinematics › Scalars vs vectors — definition, examples

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Step 0 — Trigonometry ka woh ek piece jo humein chahiye: cosine

HUM ISSE AB KYUN INTRODUCE KAR RAHE HAIN. Humari derivation ek slanted arrow ko ek seedhi baseline se compare karti hai. Woh ek number jo "yeh angle us baseline ke along kitna jhukta hai" capture karta hai, woh cosine hai. Hum isse ek baar yahan meet karte hain taaki baad mein yeh kabhi surprise na kare.

PICTURE. Ek right triangle jisme angle $\alpha$, uski adjacent side aur hypotenuse marked hain, aur neeche teen landmark values hain.

Recall Cosine negative kyun ho sakta hai

$90^{\circ }$ ke baad direction baseline ke along backward jhukna shuru ho jaati hai. "Forward" positive tha, toh "backward" negative hai. Yahi sign hai joh minus-to-plus flip ka poora engine hai joh tum Step 5 mein dekhoge.

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Step 1 — Vector kya hai, arrow ke roop mein?

KYA. Hum apni do quantities lете hain — unhe $\vec{A}$ aur $\vec{B}$ kaho — aur har ek ko ek arrow ki tarah draw karte hain.

KYUN. Kyunki poori difficulty (parent ke walking problem se) yeh hai ki direction answer change kar deta hai. Ek arrow woh ek picture hai jo dono size aur direction ek saath store karta hai, isliye yeh quantity represent karne ka honest tarika hai.

PICTURE. Do arrows jo ek hi dot se start hote hain (unke tails touch karte hain). Unke beech ka angle $\theta$ hai (Greek letter "theta", sirf us opening angle ka ek naam).

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Step 2 — Do vectors ko "add" karne ka matlab kya hai?

KYA. Add karne ke liye, hum $\vec{B}$ ko slide karte hain bina use ghoomaye, jab tak uski tail $\vec{A}$ ke head par na baith jaaye. Phir hum ek naya arrow $\vec{R}$ draw karte hain $\vec{A}$ ki tail se moved $\vec{B}$ ke head tak. Uski length $R=|\vec{R}|$ hai.

KYUN. "$\vec{A}$ karo, phir $\vec{B}$ ke saath continue karo" yehi hai head-to-tail placement — tum pehla walk khatam karte ho aur doosra wahan se shuru karte ho. Seedha ghar jaane wala arrow net effect hai, resultant $\vec{R}$.

PICTURE. Head-to-tail chain ek triangle banaati hai; $\vec{R}$ use close karta hai.

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Step 3 — Triangle ke andar ka angle $\theta$ NAHI hai

KYA. Head-to-tail triangle mein, join par ka angle (jahan $\vec{A}$ ka head $\vec{B}$ ki tail se milta hai) $180^{\circ }-\theta$ hai, $\theta$ nahi.

KYUN. Jab humne $\vec{B}$ ko $\vec{A}$ ke head par slide kiya, humne use same direction mein rakhha. Lekin ab $\vec{A}$ us corner mein aakar milta hai aur $\vec{B}$ us corner se bahar jaata hai. Do arrows ek straight-through path ke along lie karte hain, toh interior corner original opening angle ka supplement hai: yeh straight $180^{\circ }$ ka bacha hua hissa fill karta hai.

PICTURE. Corner ka zoom: original $\theta$ aur uska supplement $180^{\circ }-\theta$ saath dikhaye gaye hain ek seedhi line mein add hote hue.

Yeh ek fact — supplement, $\theta$ nahi — woh reason hai ki derivation ke beech mein ek minus appear hota hai aur phir plus mein flip ho jaata hai. Iske liye watch karo.

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Step 4 — Hum Law of Cosines kyun use karte hain (Pythagoras kyun nahi)

KYA. Hum apne triangle ke pieces ko law se match karne ke liye name karte hain:

• side $a=A$, side $b=B$,
• side $c=R$ (woh unknown jo hum chahte hain),
• angle $C=180^{\circ }-\theta$ (Step 3 se interior corner, jo $R$ ke opposite baitha hai).

KYUN. Formula us angle ki maang karta hai jo us side ke opposite ho jise tum solve kar rahe ho. $\vec{R}$ triangle ko close karta hai, toh woh corner jo iske samne hai — Step 3 se $180^{\circ }-\theta$ — exactly woh angle hai jo plug in karna hai.

PICTURE. Triangle ko $a,b,c,C$ se relabel kiya gaya hai jo $A,B,R,(180^{\circ }-\theta )$ par map karta hai.

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Step 5 — Substitute karo, aur minus ko plus mein badlo

KYA. Apne labels ko Law of Cosines mein daalo. Yahan $R$ abhi bhi length $|\vec{R}|$ mean karta hai: $R^2=A^2+B^2-2AB\cos (180^{\circ }-\theta ).$

Har term, exactly wahan jahan woh rehti hai:

• $R^2$ ::: resultant length $|\vec{R}|$ ka square (jise hum solve kar rahe hain)
• $A^2$, $B^2$ ::: "Pythagoras part" — jo tum paate agar triangle mein right angle hota
• $-2AB\cos (180^{\circ }-\theta )$ ::: correction, abhi bhi Step 3 se supplement angle rakhe hue

NEXT MOVE KYUN. Woh $\cos (180^{\circ }-\theta )$ awkward hai — yeh interior angle use karta hai, lekin hum answer vectors ke original angle $\theta$ ke terms mein chahte hain. Hum Step 0 mein flag ki gayi identity use karte hain translate karne ke liye: $\cos (180^{\circ }-\theta )=-\cos \theta .$

KYA (khatam karo). Identity substitute karo: $R^2=A^2+B^2-2AB(-\cos \theta )=A^2+B^2+2AB\cos \theta .$

Minus times minus plus ban jaata hai. Square root lo (lengths kabhi negative nahi hoti, toh hum $+$ root rakhte hain), aur yaad karo $R\equiv |\vec{R}|$: $\boxed{|\vec{R}|=\sqrt{A^2+B^2+2AB\cos \theta }}$

PICTURE. Sign-flip correction term ki ek reflection ke roop mein dikhaya gaya, "$-$ meets $-$ becomes $+$".

Yeh parent ka formula hai — ab sirf stated nahi, ab earned hai.

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Step 6 — Har case check karo (formula tumhe kabhi surprise nahi karna chahiye)

Ek accha formula saare inputs cover karta hai. Chalo isme har angle daalo aur dekho yeh kaisa behave karta hai. Poore mein, $R=|\vec{R}|$.

Case $\theta =0^{\circ }$ (same direction). $\cos 0^{\circ }=1$: $R=\sqrt{A^2+B^2+2AB}=\sqrt{(A+B{)}^2}=A+B.$ Arrows ek line mein stack hote hain — maximum length, plain arithmetic. Yahi reason hai ki aligned vectors "scalars ki tarah act karte hain".

Case $\theta =90^{\circ }$ (perpendicular). $\cos 90^{\circ }=0$: $R=\sqrt{A^2+B^2}.$ Correction khatam ho jaata hai; hum pure Pythagoras par wapas aa jaate hain. (Parent mein Example 1 ka $3,4,5$ triangle yahi hai.)

Case $\theta =180^{\circ }$ (opposite directions). $\cos 180^{\circ }=-1$: $R=\sqrt{A^2+B^2-2AB}=\sqrt{(A-B{)}^2}=|A-B|.$ Arrows ladai karte hain; result unka difference hota hai — minimum length. Agar $A=B$ toh woh $0$ tak cancel ho jaate hain.

Case $\theta$ beech mein (jaise $120^{\circ }$). $\cos 120^{\circ }=-\frac{1}{2}$, giving $R=\sqrt{A^2+B^2-AB}$ — ek value smoothly extremes ke beech.

PICTURE. Ek figure jo $R$ ko $A+B$ se ghatat hue $|A-B|$ tak dikhata hai jaise $\theta$ $0^{\circ }$ se $180^{\circ }$ tak khulta hai.

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Ek-picture summary

Is page ka har idea, ek frame mein stacked: do arrows tail-to-tail ($\theta$), slid head-to-tail (corner $180^{\circ }-\theta$), $\vec{R}$ se closed, law of cosines ke saath aur sign-flip jo ise $\sqrt{A^2+B^2+2AB\cos \theta }$ mein turn kar deta hai, plus neeche min/max dial.

Recall Feynman retelling — poora walk plain words mein

Pehle, ek chota tool: ek right triangle par, ek angle ka cosine sirf "slanted side kitni forward lean karti hai" hai — ek number $-1$ aur $1$ ke beech, negative jab yeh backward lean kare. Ise apni pocket mein rakho. Ab do arrows imagine karo jo same jagah se nikal rahe hain. Unhe add karne ke liye, main doosra arrow uthata hoon, use same direction mein point karta rehta hoon, aur uski tail pehle ki tip par chipka deta hoon — jaise ek walk lena, phir wahan se doosri walk karna jahan main ruka tha. Mere start se mere finish tak ki seedhi line answer arrow hai, aur uski length woh hai jise main $R$ keh raha hoon ($|\vec{R}|$ ka same, sirf zyada bars ke bina). Woh answer ek triangle ki teesri side hai. Ek rule hai triangle ki teesri side find karne ke liye jab tum baaki do sides aur unke beech ka corner jaante ho — law of cosines. Lekin ek twist hai: mere triangle ke andar ka corner woh angle nahi hai jisse maine shuru kiya tha; yeh woh hai jo ek seedhi line se bach jaata hai, $180^{\circ }$ minus mera angle. Jab main woh daalta hoon, cosine sign flip kar leta hai (forward backward ban jaata hai), ek minus ek minus se milta hai, aur ek plus nikalta hai. Yahi famous $+2AB\cos \theta$ hai. Phir main har angle try karta hoon. Unhe same direction mein point karo aur woh apne full sum tak stack hote hain. Unhe right angles par point karo aur yeh plain Pythagoras hai. Unhe opposite point karo aur woh apne difference tak cancel hote hain. Answer arrow sum se lamba ya difference se chota kabhi nahi ho sakta — yeh sirf smoothly un dono ke beech dial karta hai jaise main angle kholata hoon. Woh dial hi poora reason hai ki vectors scalars nahi hain.

Recall Quick self-test

Correction term $-$ (law of cosines) se $+$ (final formula) mein kyun switch hota hai? ::: Kyunki triangle ka interior angle $180^{\circ }-\theta$ hai, aur $\cos (180^{\circ }-\theta )=-\cos \theta$ sign flip kar deta hai. Kaunsa angle sabse lamba resultant deta hai? ::: $\theta =0^{\circ }$, giving $R=A+B$. Kaunsa angle plain Pythagoras recover karta hai? ::: $\theta =90^{\circ }$, kyunki $\cos 90^{\circ }=0$. Equal length ke do vectors opposite direction mein point karte hain — resultant? ::: Zero ($|A-B|$ case jisme $A=B$). $R$ aur $|\vec{R}|$ mein kya relation hai? ::: Woh same number hain — $\vec{R}$ ki length; bars optional hain jab clear ho ki hum length mean kar rahe hain.

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Connections

• Vector Addition — Triangle & Parallelogram Law — is result ka ghar; parallelogram view same triangle mirrored hai.
• Components of a Vector — same $R$ tak pahunchne ka alternative route, har arrow ko perpendicular scalar parts mein split karke.
• Distance vs Displacement — $\theta =90^{\circ }$ special case action mein.
• Speed vs Velocity — jahan $\theta =180^{\circ }$ cancellation round trip par dikhti hai.
• Dot and Cross Products — yahan ka $\cos \theta$ exactly dot product disguise mein hai.
• Units and Dimensions — $A$, $B$, $R$ har step mein same unit carry karte hain.