Cross product — formula, direction (right-hand rule), torque - area calculation: Question bank

1.1.12 · D5 · Physics › Measurement, Vectors & Kinematics › Cross product — formula, direction (right-hand rule), torque

Traps se pehle, ek shared picture jise hum point kar sakein. Do arrows tail-to-tail rakhne par ek tircha patch sweep karte hain; cross product ki length us patch ka area hai, aur uski direction woh skewer hai jo patch se seedha upar khada hai.

Figure — Cross product — formula, direction (right-hand rule), torque - area calculation

True ya false — justify karo

Har item ke liye: true/false decide karo, phir ek-sentence ka reason do. Reason hi asli cheez hai.

A \times B

ek scalar hai kyunki ye area measure karta hai.

False. Ye apni magnitude mein ek area encode karta hai, lekin poora object ek vector hai — area ke saath ek perpendicular direction bhi aata hai (woh "patch kis taraf face karta hai" wala skewer), aur woh direction zaroori hai.

Agar

∣ A \times B ∣ = 0

ho, toh kam se kam ek vector zero vector hai.

False. Ye tab bhi zero hota hai jab dono nonzero hon lekin parallel ya antiparallel hon (

θ = 0^{\circ}

18 0^{\circ}

sin θ = 0

) — ek hi line par do arrows zero width ka patch span karte hain.

A \times B

hamesha ek aisi direction mein point karta hai jo dono

A

aur

B

ke perpendicular ho.

True. Ye us poore plane ke perpendicular hota hai jise dono span karte hain, isliye us plane mein lie karne wale har vector ke perpendicular hai, including

A

aur

B

khud.

Order badalna,

B \times A

, alag length ka vector deta hai.

False. Same length (

A B sin θ

change nahi hota), opposite direction — right-hand curl reverse ho jaata hai, toh thumb flip ho jaata hai:

B \times A = - A \times B

A \times A = 0

har vector

A

ke liye.

True. Ek vector khud ke parallel hota hai,

θ = 0

, toh

sin 0 = 0

aur koi patch nahi banta — ek line ka koi area nahi hota.

Cross product tab sabse bada hota hai jab do vectors sabse zyada aligned hon.

False. Ye tab sabse bada hota hai jab dono perpendicular hon (

θ = 9 0^{\circ}

sin θ = 1

). Alignment toh dot product reward karta hai; cross product spread ko reward karta hai.

(A \times B) \cdot A = 0

hamesha.

True.

A \times B

A

ke perpendicular hai, aur perpendicular vectors ka dot product zero hota hai — ye kisi bhi computed cross product ka ek quick sanity check hai.

Cross product associative law follow karta hai:

A \times (B \times C) = (A \times B) \times C

False. Cross product associative nahi hai; dono sides generally alag-alag point karte hain kyunki parenthesised pair har baar ek alag plane define karta hai.

Error dhundho

Reasoning padho, galat step dhundho, aur fix batao.

F

bahut bada hai, toh torque

τ = r \times F

bhi bahut bada hoga."

Error direction ko ignore karta hai: agar

F

r

ke parallel hai (lever ke seedha along push karna), toh

sin θ = 0

aur

τ = 0

chahe

F

kitna bhi bada ho. Sirf perpendicular part hi cheezein turn karta hai.

"Triangle

P QR

ka area nikalne ke liye, main origin

O

use karke

\frac{1}{2} ∣ O P \times O Q ∣

leta hoon."

Galat tails — do edge vectors ko triangle ka ek vertex share karna chahiye, jaise

P

P Q

aur

P R

. Unrelated point

O

se vectors galat parallelogram span karte hain.

"Determinant ko left to right expand karte waqt,

\hat{j}

term hai

A_{x} B_{z} - A_{z} B_{x}

Cofactor minus sign missing hai. Pattern

+, -, +

hai, toh

\hat{j}

term hai

- (A_{x} B_{z} - A_{z} B_{x}) = A_{z} B_{x} - A_{x} B_{z}

\hat{j} \times \hat{i} = \hat{k}

kyunki

\hat{i} \times \hat{j} = \hat{k}

Order matter karta hai — order reverse karne par sign flip ho jaata hai, toh

\hat{j} \times \hat{i} = - \hat{k}

. Cycle

\hat{i} \to \hat{j} \to \hat{k}

sirf aage jaate waqt

+

hota hai.

"Torque

10 \hat{k}

aaya, toh object

+ \hat{k}

direction mein rotate karega."

Torque ek rotation direction nahi hai; ye axis batata hai.

+ \hat{k}

(page se bahar) matlab right-hand rule se page mein counter-clockwise spin — arrow spin-axis hai, motion ka path nahi.

"Kyunki

A \times B

mein

sin θ

hai aur

sin

obtuse angles ke liye negative ho sakta hai, magnitude negative ho sakti hai."

Do vectors ke beech ke angle ke liye hum hamesha

0^{\circ} \leq θ \leq 18 0^{\circ}

lete hain, jahan

sin θ \geq 0

, toh magnitude kabhi negative nahi hoti. Sign information direction mein rehti hai, length mein nahi.

A \times B

aur

B \times A

dono same parallelogram describe karte hain, toh ye same vector hain."

Dono same patch describe karte hain (same area) lekin skewer uske opposite faces ki taraf point karta hai — same magnitude, opposite direction, isliye ye minus sign se alag hain.

Why questions

"What" nahi, "why" ka jawab do.

Cross product vector kyun return karta hai jabki dot product number return karta hai?

Kyunki rotation aur orientation ko space mein ek direction chahiye (axis kis taraf point karta hai, surface kis taraf face karti hai), aur ek plain number wo carry nahi kar sakta — ek perpendicular vector kar sakta hai. Scalar counterpart ke liye dekho Dot product — formula, projection, work calculation.

Parallelogram ki height

B sin θ

kyun hai,

B cos θ

kyun nahi?

Height base

A

ke perpendicular measure hoti hai;

B

ka perpendicular component

B sin θ

hai, jabki

B cos θ

woh part hai jo $A$ ke along hai, jo width mein kuch nahi jodta.

Sirf force ka perpendicular part hi torque kyun create karta hai?

Lever arm ke along point karne wala force pivot ki taraf ya usse door push karta hai aur use swing nahi kar sakta; sirf sideways (

⊥

) component use twist karta hai — ye exactly door-at-the-hinge wali picture hai. Dekho Torque and rotational equilibrium.

Determinant ke middle term mein minus sign kyun hota hai?

Alternating

+, -, +

cofactor pattern anti-commutativity ka algebraic fingerprint hai; ye guarantee karta hai ki

A

aur

B

rows swap karne par poore result ka sign flip ho. Isi tarah built hai jaise Determinants — 3×3 expansion mein.

A \times B

sirf ek vector ke nahi, poore plane ke perpendicular kyun guaranteed hai?

Magnetic-force aur area interpretations ko poore patch ke liye ek single unambiguous "facing" direction chahiye; plane ke perpendicular hone se ye ek saath us mein har vector ke perpendicular ho jaata hai. Isliye Magnetic force — F = qv × B ise use karta hai.

Hum kyun kehte hain ki cross product "tab maximal hai jab dot product zero ho"?

Dono

sin θ

aur

cos θ

use karte hain, jo complementary hain: jab vectors perpendicular hon,

sin θ = 1

(cross max hai) aur

cos θ = 0

(dot zero hai), aur parallel hone par ulta.

Do vectors ko curl karne se pehle tail share kyun karni chahiye?

Angle

θ

aur plane tabhi well-defined hote hain jab dono arrows same point se start karein; shared tail se

A

B

tak ka sweep unambiguous hai, jo magnitude aur direction dono fix karta hai.

Edge cases

Boundary aur degenerate inputs — woh scenarios jo formulas quietly assume kar lete hain.

A \times B

kya hoga jab

θ = 0^{\circ}

(parallel) ho?

Zero vector:

sin 0 = 0

se zero magnitude milti hai, aur koi plane nahi banta direction define karne ke liye — "patch" ek line mein collapse ho gaya.

A \times B

kya hoga jab

θ = 18 0^{\circ}

(antiparallel) ho?

Yahan bhi zero vector:

sin 18 0^{\circ} = 0

. Arrows ek hi line par opposite directions mein hain, phir bhi koi area span nahi karte.

Kya hoga agar ek input zero vector ho, jaise

A = 0

Result

0

hoga — koi arrow nahi hai patch span karne ke liye, toh magnitude (

A = 0

) aur direction dono undefined-but-zero hain.

Exactly

θ = 9 0^{\circ}

par, magnitude aur direction kaisi behave karti hain?

Magnitude apna maximum

A B

hit karti hai (kyunki

sin 9 0^{\circ} = 1

), aur direction cleanly dono ke perpendicular hoti hai — ye "sabse clean" cross product hai, reference case.

Agar

A

aur

B

3D mein hain lekin dono

x y

-plane mein lie karte hain, toh

A \times B

kahan point karega?

Seedha

\pm \hat{k}

ke along (plane se bahar ya andar), kyunki poori

x y

-plane ke perpendicular direction sirf

z

-axis hai — sign right-hand rule se set hota hai.

Angular momentum

L = r \times p

ke liye,

L

kya hoga jab particle seedha origin ki taraf move kare?

Zero, kyunki

r

aur

p

tab parallel hote hain (

θ = 0

), toh koi rotational "spread" nahi hota. Dekho Angular momentum — L = r × p.

Kya do alag vector pairs bilkul same cross product vector de sakti hain?

Haan — koi bhi pair jo same orientation ke saath same plane mein same-area parallelogram span kare, identical result deti hai; cross product apne inputs ko uniquely determine nahi karta.

Recall Jaane se pehle ek-line self-test

Cross product: length hai area of the parallelogram ( $A B sin θ$ ), direction hai right-hand skewer perpendicular to the plane, aur ye parallel vectors ke liye vanish hota hai aur perpendicular ones ke liye peak karta hai.

Connections

Cross product — formula, direction (right-hand rule), torque - area calculation (parent)
Dot product — formula, projection, work calculation
Vectors — addition, components, unit vectors
Torque and rotational equilibrium
Angular momentum — L = r × p
Magnetic force — F = qv × B
Determinants — 3×3 expansion