Vector addition — triangle law, parallelogram law: Question bank

1.1.8 · D5 · Physics › Measurement, Vectors & Kinematics › Vector addition — triangle law, parallelogram law

Do anchors jis par aap throughout tike rahenge:

Har formula mein angle, $θ$ , woh angle hai jo do vectors ke beech tail-to-tail measure hota hai — dono arrows ek shared point se shuru hote hain.
Do master formulas: $R = A^{2} + B^{2} + 2 A B cos θ$ (magnitude) aur $tan α = \frac{B sin θ}{A + B cos θ}$ (direction, $A$ se angle).

True or false — justify

Do vectors jinki length 3 aur 4 hai, unka resultant kabhi 8 length ka nahi ho sakta.

True. Resultant ki maximum limit

A + B = 7

hai (jo sirf

θ = 0^{\circ}

par milti hai), isliye

8

impossible hai — yeh allowed band

[∣ A - B ∣, A + B] = [1, 7]

se bahar hai.

3 aur 4 length ke do vectors ek resultant jo length 1 ka ho, produce kar sakte hain.

True. Yahi minimum hai,

∣ A - B ∣ = 1

, jo tab milta hai jab dono exactly opposite directions mein point karein (

θ = 18 0^{\circ}

Agar

A + B

ki magnitude

A - B

ke barabar hai, to

A

aur

B

perpendicular hain.

True. Dono ko square karke,

A^{2} + B^{2} + 2 A B cos θ = A^{2} + B^{2} - 2 A B cos θ

cos θ = 0

milta hai, matlab

θ = 9 0^{\circ}

(yeh baat Scalar (dot) product se bhi dekh sakte hain, kyunki

A \cdot B = 0

Do vectors ka resultant hamesha unke beech hota hai (jis angle se woh milte hain, uske andar).

True (us angle ke liye jo woh tail-to-tail subtend karte hain). Parallelogram ka diagonal apni do adjacent sides ke beech baitha rehta hai, isliye resultant ki direction hamesha

A

aur

B

ke beech squeeze rehti hai.

Do equal-length vectors ke liye, resultant unke beech ke angle ko bisect karta hai.

True. Symmetry ki wajah se: dono identical vectors ko swap karne se figure unchanged rehta hai, isliye resultant kisi ek ko favour nahi kar sakta — use angle bisector par hi rehna padega.

Parallelogram law, triangle law se unhi do vectors ke liye alag answer deta hai.

False. Yeh dono same operation ke alag-alag disguise hain; parallelogram ki opposite side

B

ke barabar hai, isliye uska diagonal exactly

A

aur

B

ke head-to-tail triangle ko close karta hai.

Vector addition commutative hai:

A + B = B + A

True. Chahe aap journey

A

phir

B

chalein, ya

B

phir

A

, aap parallelogram ke same corner par pahunchte hain — same endpoint, same resultant.

Agar magnitudes match karein to aap

5 N

force ko

5 m/s

velocity mein add kar sakte hain.

False. Sirf like quantities add hoti hain; ek force aur ek velocity alag-alag unit-worlds mein rehte hain, isliye unka sum meaningless hai chahe numbers kuch bhi hon.

Agar

θ

0^{\circ}

18 0^{\circ}

tak badhta hai, to resultant magnitude poore raaste ghat'ta rehta hai.

True.

R^{2} = A^{2} + B^{2} + 2 A B cos θ

mein

cos θ

monotonically

+ 1

- 1

tak

[0^{\circ}, 18 0^{\circ}]

par girta hai, aur

R

ko bhi apne saath kheenchta hai.

Teen vectors tab hi zero vector mein add ho sakte hain jab teeno ki magnitude equal ho.

False. Koi bhi teen vectors jo ek closed head-to-tail triangle banate hain, zero mein sum hote hain — sides equal nahi chahiye, bas close ho sakne chahiye.

Why questions

Resultant hamesha

A + B

se chhota kyun hota hai jab tak vectors parallel na hon?

Kyunki head-to-tail path join par muda hota hai; ek muda hua path shuru se aakhir tak hamesha do seedhe segments se chhhota hota hai (triangle inequality). Sirf

0^{\circ}

ka bend poori length

A + B

ko preserve karta hai.

Magnitude formula mein

cos θ

—

sin θ

nahi — kyun aata hai?

cos θ

measure karta hai ki

B

ka kitna hissa

A

ke along point karta hai; wahi aligned part resultant ko stretch karta hai. Yahi exactly woh piece hai jo Scalar (dot) product

A \cdot B = A B cos θ

isolate karta hai.

θ

tail-to-tail kyun measure hona chahiye, head-to-tail nahi?

Formulas parallelogram se derive hue hain jisme dono tails ek point par hain; arrows jis "opening" angle se us point se nikalte hain, wahi geometry control karta hai. Head-to-tail measurement supplement,

18 0^{\circ} - θ

, deta hai, aur cross term ka sign flip kar deta hai.

Hum

B

(B cos θ, B sin θ)

mein resolve karke components alag-alag kyun add kar sakte hain?

Kyunki

x

aur

y

ke along displacement independent journeys hain — ek trip ka eastward part uske northward part se unaffected hai (dekho Resolution of a vector into components aur Vectors — components and unit vectors).

Direction formula

R_{y} / R_{x}

use kyun karta hai, input angles ka koi average nahi?

Resultant ka tilt wahan fix hota hai jahan uska head land karta hai, jiska vertical rise

R_{y} = B sin θ

aur horizontal run

R_{x} = A + B cos θ

milke

tan α

define karte hain. Input directions ka average vector ki lengths ko ignore karta hai, isliye jab

A \neq = B

ho to fail ho jaata hai.

Relative velocity (nadi paar karna) vector-addition problem kyun hai, subtraction nahi?

Ground ke relative boat ki velocity, boat-in-water velocity plus water-relative-to-ground velocity hai — do journeys head-to-tail chain hoti hain. Dekho Relative velocity ki kaise current resultant heading ko shift karta hai.

Edge cases

Jab do vectors mein se ek zero vector ho to resultant kya hoga?

Sum doosra vector unchanged hoga:

A + 0 = A

. Zero vector koi length ki journey nahi hai, isliye kuch add nahi karta aur result ko shift karne ke liye koi direction nahi hai.

Jab

A + B cos θ = 0

ho to direction formula ka kya hoga?

Denominator zero ho jaata hai,

tan α \to \infty

, matlab

α = 9 0^{\circ}

: resultant exactly

A

ke perpendicular hai. Yeh tab hota hai jab

B

ka backward pull

A

ki poori length base ke along cancel kar de.

Do equal vectors exactly opposite directions mein point kar rahe hain (

θ = 18 0^{\circ}

A = B

). Resultant kya hai?

Zero vector.

R = ∣ A - B ∣ = 0

; woh perfectly cancelling pair hain, aur direction undefined hai kyunki koi leftover journey nahi hai.

Jab

θ \to 0^{\circ}

ho to parallelogram kis cheez mein degenerate ho jaata hai?

Yeh ek line segment mein collapse ho jaata hai — dono vectors ek doosre ke upar aate hain, "parallelogram" ki zero width ho jaati hai, aur uska diagonal straight sum

A + B

ban jaata hai jo same direction mein point karta hai.

Kya do nonzero vectors ka resultant dono ke opposite point kar sakta hai?

Nahi. Resultant hamesha do vectors ke beech ke wedge mein rehta hai, isliye woh kabhi us region se bahar nahi ja sakta — dono ke opposite point karna to bahut door ki baat hai.

Agar dono vectors ki same magnitude ho aur

θ = 12 0^{\circ}

ho, to resultant ki length kya hogi?

R = A^{2} + A^{2} + 2 A^{2} cos 12 0^{\circ} = 2 A^{2} - A^{2} = A

— resultant ek single vector ki length ke barabar hai, aur symmetry se

12 0^{\circ}

angle ko

6 0^{\circ}

par bisect karta hai har ek se.

Recall Ek-line survival kit

Angle tail-to-tail hota hai · $R \in [∣ A - B ∣, A + B]$ · $R = A^{2} + B^{2} + 2 A B cos θ$ · head-to-tail = sum, tips-joined = difference · diagonal = resultant, sides = vectors hain.

Connections

Vector addition — triangle law, parallelogram law (parent)
Subtraction of vectors and the difference vector
Resolution of a vector into components
Vectors — components and unit vectors
Relative velocity
Law of cosines
Scalar (dot) product

Question bank — Vector addition — triangle law, parallelogram law