Scalars vs vectors — definition, examples: Question bank

1.1.6 · D5 · Physics › Measurement, Vectors & Kinematics › Scalars vs vectors — definition, examples

Figure — Scalars vs vectors — definition, examples

Formula Woh ek formula jis par har item tika hai — picture se banaya gaya

$A$ aur $B$ ko tail-to-tail rakho aur parallelogram complete karo; diagonal $R$ hai. Uski length yeh hai: $∣ R ∣ = A^{2} + B^{2} + 2 A B cos θ .$ Yeh $A + B$ kyun nahi? $B$ ko slide karo taaki uski tail $A$ ki tip par baith jaaye (yahi triangle law hai — neeche ki figure slide dikhati hai). Ab $A$ , slid $B$ , aur closing arrow $R$ ek triangle banate hain. Law of cosines — "do sides aur unke beech ka angle se teesri side nikalo" ka sahi tool — us corner ke interior angle ki zaroorat hai jahan $A$ khatam hota hai aur $B$ shuru hota hai.

Woh interior angle $(18 0^{\circ} - θ)$ kyun hai, $θ$ nahi: original angle $θ$ tail-to-tail measure hota hai. Lekin triangle mein, $A$ us corner par head-first aata hai aur $B$ wahan se tail-first jaata hai — do arrows nose-to-tail rakhe hain, isliye arrow directions tail-to-tail picture ke relative flip ho gayi hain. Corner ka opening isliye $θ$ ka supplement hai, yaani $18 0^{\circ} - θ$ . Figure mein dono $θ$ (blue, tail-to-tail) aur uska supplement (plum, triangle ke andar) mark hain taaki tum dekh sako ki woh straight line mein add hote hain. Plug in karne par: $∣ R ∣^{2} = A^{2} + B^{2} - 2 A B cos (18 0^{\circ} - θ) = A^{2} + B^{2} + 2 A B cos θ,$ kyunki $cos (18 0^{\circ} - θ) = - cos θ$ . $+ 2 A B cos θ$ wala term exactly "kitna $B$ , $A$ ke saath align karta hai" yeh batata hai.

True or false — justify karo

Temperature ek scalar hai kyunki yeh ek number aur ek unit se poori tarah describe ho jaati hai.

True. "27 °C" ko koi direction nahi chahiye; tum kabhi do temperatures ko tail-to-tail rakh ke diagonal nahi nikalte, isliye koi parallelogram law apply nahi hota — yeh pure scalar hai.

Circuit wire mein electric current ek vector hai kyunki yeh ek definite direction mein flow karta hai.

False (circuit quantity ke roop mein). Ek junction par, currents arithmetically add hote hain — tum literally charges per second count karte ho andar aur bahar,

I_{1} + I_{2} = I_{3}

(Kirchhoff), koi diagonal nahi, koi

cos θ

nahi. Isliye circuit current

I

ek scalar hai. (Dhyan raho: related current density

J

, "ek point par current per unit area", electromagnetism mein ek genuine vector field hai — lekin woh ek wire ke single number

I

se alag quantity hai. Trap dono ko confuse karta hai.)

Kisi vector ki magnitude negative ho sakti hai.

False. Magnitude ek length hai (

se nikali jaati hai), isliye yeh

\geq 0

hoti hai. Vector par minus sign uski direction flip karta hai, uski length ko negative nahi banata.

Agar do vectors ki magnitude equal ho, toh unka sum dono mein se kisi ek se bada hona chahiye.

False. Unhe

θ = 18 0^{\circ}

par tail-to-tail rakho (opposite direction mein): diagonal collapse ho jaata hai,

A^{2} + A^{2} + 2 A^{2} cos 18 0^{\circ} = 2 A^{2} - 2 A^{2} = 0

. Sum zero hai, dono se chota.

Distance travelled, displacement ki magnitude se choti ho sakti hai.

False. Distance total path length hai aur displacement seedha-seedha shortcut hai, isliye distance

\geq ∣ displacement ∣

hamesha — barabar sirf tab jab path straight line ho aur koi reversal na ho.

Ek poore lap par, average speed zero hoti hai.

False. Speed distance use karta hai (poore lap ki length, jo nonzero hai), isliye yeh nonzero hai. Average velocity zero hoti hai, kyunki displacement zero hai.

Ek scalar aur vector ko add karna, jaise "5 kg + ek force", ek meaningful operation hai.

False. Alag categories (aur aksar alag dimensions, dekho Units and Dimensions) ko add nahi kiya ja sakta; sirf same-type quantities combine hoti hain.

Do vectors hamesha ek aisa resultant produce karte hain jiski magnitude

∣ A - B ∣

aur

A + B

ke beech hoti hai.

True. Jaise

θ

0^{\circ} \to 18 0^{\circ}

tak jaata hai,

cos θ

+ 1 \to - 1

tak jaata hai, isliye

∣ R ∣

continuously maximum

A + B

(arrows aligned, diagonal poori tarah stretched) se minimum

∣ A - B ∣

(arrows opposed, diagonal shrunk) tak sweep karta hai.

Error dhundho

"3 km East aur 4 km North ka displacement milane par 7 km milta hai."

Error yeh hai ki non-collinear vectors ki magnitudes arithmetically add kar di gayi hain. Unhe

θ = 9 0^{\circ}

par tail-to-tail draw karo: diagonal hai

∣ R ∣ = 3^{2} + 4^{2} + 2 \cdot 3 \cdot 4 cos 9 0^{\circ} = 9 + 16 = 5

km (

cos 9 0^{\circ} = 0

cross term ko khatam kar deta hai), na ki

7

"Speed sirf bina arrow wali velocity hai, isliye unka number hamesha same hoga."

Yeh sirf straight, non-reversing path par agree karte hain. Kisi bhi curved ya back-and-forth motion par distance shortcut se zyada hoti hai, isliye average speed

>

average velocity ki magnitude (tum scalar speed ko

∣ average velocity ∣

se compare karo, na ki signed vector se). Dekho Speed vs Velocity.

"Kyunki force ek vector hai,

6 N + 8 N

kabhi

14

N nahi ho sakta."

Vectors scalar ki tarah add ho sakte hain jab aligned hon.

θ = 0^{\circ}

set karo taaki

cos 0^{\circ} = 1

ho; phir diagonal seedha dono arrows ke saath lie karta hai:

36 + 64 + 2 \cdot 6 \cdot 8 \cdot 1 = 196 = 14

N. Parallelogram rule us special case mein plain addition mein reduce ho jaata hai.

"Work mein direction hai kyunki force mein direction hai, isliye work ek vector hai."

Work force aur displacement ka dot product hai — tum do lengths aur unke beech ke angle ka cosine multiply karte ho, aur length-times-length-times-number sirf ek plain number hai (Dot and Cross Products). Work ek scalar hai — yeh positive ya negative ho sakta hai, lekin woh sign energy in vs out mean karta hai, koi direction nahi.

"Negative charge ek vector hai kyunki sign kisi direction mein point karta hai."

Charge par sign ek type/polarity label hai (number line par zero se neeche), koi spatial direction nahi. Do charges

q_{1} + q_{2}

ki tarah add hote hain bina kisi angle ke, isliye charge ek scalar hai.

"Kyunki velocity ek vector hai, uski magnitude (speed) bhi ek vector hai."

Kisi vector ki magnitude lene se direction strip ho jaati hai aur ek scalar milta hai. Speed scalar

∣ v ∣

hai.

Why questions

"Direction hona" kisi quantity ko vector bolne ke liye kaafi kyun nahi hai?

Kyunki circuit current jaisi counted quantities mein bhi direction hoti hai phir bhi woh arithmetically add hoti hain (

I_{1} + I_{2}

, koi diagonal nahi). Decisive test yeh hai ki kya woh tail-to-tail rakhe jaane par triangle/parallelogram law follow karte hain.

Resultant formula mein sirf

A + B

ki jagah

cos θ

kyun hai?

Kyunki resultant triangle ki teesri side hai, aur law of cosines ko baaki do sides ke beech ka angle chahiye.

cos θ

term encode karta hai ki $B$ ka kitna hissa $A$ ke saath align karta hai: jab woh align hote hain toh poora add hota hai (

cos 0^{\circ} = 1

), jab perpendicular hote hain toh kuch contribute nahi karta (

cos 9 0^{\circ} = 0

Ek trip mein vector ki average value zero kyun ho sakti hai jab "clearly kuch hua tha"?

Directed quantities cancel kar sakti hain: equal-and-opposite contributions (tail-to-tail at

18 0^{\circ}

) zero sum karte hain. Round trip ke forward aur return displacements annihilate ho jaate hain, net zero dete hain.

Hum vectors ko components (scalar pieces) mein kyun todh dete hain?

Socho arrow ki seedhi shadow horizontal aur vertical axes par giraa rahe ho — har shadow ek signed number hai, aur ek axis par numbers simply add ho jaate hain. Isse hard parallelogram addition, aasaan per-axis sums mein badal jaati hai, woh trick jo Components of a Vector mein expand hoti hai.

Distance ka scalar hona yeh kyun guarantee karta hai ki woh chalte rehne par kabhi decrease nahi kar sakta?

Distance path length accumulate karta hai, jo hamesha positive amount hoti hai, isliye woh sirf badhta hi hai. Displacement shrink kar sakta hai kyunki directed quantity reverse kar sakti hai.

Edge cases

Do equal vectors

A

θ = 18 0^{\circ}

par resultant kya hai?

Zero.

A^{2} + A^{2} + 2 A^{2} cos 18 0^{\circ} = 2 A^{2} - 2 A^{2} = 0

— minimum case, full cancellation (arrows opposed, diagonal vanish ho jaata hai).

Jab do vectors perpendicular hon (

θ = 9 0^{\circ}

) toh

∣ R ∣

kya hoga?

Cross term mar jaata hai kyunki

cos 9 0^{\circ} = 0

, aur clean Pythagorean diagonal milta hai

∣ R ∣ = A^{2} + B^{2}

— jaise

A = 3

B = 4

deta hai

5

Kya zero magnitude wala vector phir bhi vector hai?

Haan — zero vector ki magnitude

0

aur undefined/arbitrary direction hoti hai, lekin yeh vector category mein rehta hai aur additive identity hai (

A + 0 = A

Kya ek scalar negative ho sakta hai aur phir bhi scalar reh sakta hai?

Haan. Temperature

- 5

°C ya potential

- 3

V scalars hain jinke minus sign ka matlab "reference se neeche" hai, koi spatial direction nahi.

Agar do vectors anti-parallel hain (

θ = 18 0^{\circ}

) aur

A = 8

B = 5

hai, toh

∣ R ∣

kya hai?

∣ R ∣ = 8^{2} + 5^{2} + 2 \cdot 8 \cdot 5 cos 18 0^{\circ} = 64 + 25 - 80 = 9 = 3

, bade vector

A

ki direction mein. Yeh resultant range ki lower bound

∣ A - B ∣

hai.

Agar ek vector ki magnitude zero ho toh

∣ R ∣ = A^{2} + B^{2} + 2 A B cos θ

ka kya hoga?

Yeh

A^{2} = A

par collapse ho jaata hai: zero vector add karne se doosra unchanged rehta hai, jaise expected hai.

Kya average speed kabhi average velocity ki magnitude ke barabar hoti hai?

Haan — sirf tab jab motion straight line mein bina reverse kiye ho, taaki distance

= ∣ displacement ∣

aur dono ratios ek ho jaayein.

Recall Har trap ki ek-line summary

Inme sabka pattern yeh hai: direction ek symptom hai, addition rule diagnosis hai. Puchho "kya in dono ko, tail-to-tail rakh ke, parallelogram law se combine kiya jaata hai?" — agar haan, vector; agar woh sirf counted numbers ki tarah add hote hain, toh scalar.

Connections

Scalars vs vectors — definition, examples — parent note jise yeh traps drill karte hain.
Vector Addition — Triangle & Parallelogram Law — $A^{2} + B^{2} + 2 A B cos θ$ rule ka source.
Distance vs Displacement — path-length vs shortcut par trap set.
Speed vs Velocity — round-trip cancellation traps.
Components of a Vector — scalar components addition ko aasaan kyun banate hain.
Dot and Cross Products — work (dot product) scalar kyun hai.
Units and Dimensions — unlike quantities ab bhi add nahi ho sakti.