Cross product — formula, geometric meaning (area), right-hand rule: Question bank

4.5.3 · D5 · Maths › Linear Algebra (Full) › Cross product — formula, geometric meaning (area), right-han

Prerequisites jinke saath comfortable hona chahiye: parent note Cross Product, plus Dot Product, Determinants, aur Orthogonality.

True or false — justify

The cross product

a \times b

sirf

R^{3}

ke vectors ke liye defined hai.

True — vector-valued cross product 3D ke liye special hai; 2D ya higher mein perpendicular direction unique nahi hoti ya single vector nahi hoti.

a \times b = b \times a

sabhi vectors ke liye.

False — ye anticommutative hai:

b \times a = - (a \times b)

; sirf magnitude (area) symmetric hai, direction flip ho jaati hai.

Agar

a \times b = 0

, toh ya toh

a = 0

hai ya

b = 0

False — ye tab bhi zero hota hai jab

a

aur

b

parallel hon (ya antiparallel), kyunki

sin θ = 0

se zero area milta hai chahe dono vectors nonzero hon.

a \times b

hamesha

a + b

ke perpendicular hota hai.

True — kyunki ye

a

aur

b

dono ke perpendicular hai, isliye unke kisi bhi linear combination ke bhi perpendicular hai, including unka sum (dekho Orthogonality).

Magnitude

∣ a \times b ∣

kabhi

∣ a ∣ ∣ b ∣

se bada ho sakta hai.

False —

∣ a ∣∣ b ∣ sin θ

ka maximum

∣ a ∣∣ b ∣

hai, jo exactly

θ = 9 0^{\circ}

par milta hai;

sin θ \leq 1

hamesha.

a \times (b \times c)

equals

(a \times b) \times c

False — cross product associative nahi hai; regrouping se generally result badal jaata hai, isliye parentheses yahan genuinely matter karte hain.

Cross product ka order swap karne se reported area badal jaati hai.

False — swap karne se sirf direction reverse hoti hai; length

∣ a ∣∣ b ∣ sin θ

unchanged rehti hai, isliye reported area identical hoti hai.

a \times a = 0

har vector

a

ke liye.

True — ek vector apne aap ke parallel hota hai, angle

0

, isliye

sin 0 = 0

aur koi parallelogram nahi (zero area).

Identity

∣ a \times b ∣^{2} + (a \cdot b)^{2} = ∣ a ∣^{2} ∣ b ∣^{2}

sabhi

a, b

ke liye hold karti hai.

True — ye Lagrange's identity hai; ye

sin^{2} θ + cos^{2} θ = 1

ko package karta hai aur Dot Product aur cross product ko cleanly link karta hai.

Spot the error

"Cross product ek number output karta hai jo measure karta hai ki do vectors kitna align karte hain."

Do jagah galat hai: ye ek vector output karta hai, aur ye measure karta hai ki vectors kitna align karne mein fail karte hain (perpendicularity/area), alignment nahi — woh dot product ka kaam hai.

a \times b

ka middle component

a_{1} b_{3} - a_{3} b_{1}

hai."

Sign error — middle component

a_{3} b_{1} - a_{1} b_{3}

hai; determinant expansion mein

\hat{j}

cofactor minus sign carry karta hai (pattern

+, -, +

"Kyunki area symmetric hai,

a \times b

aur

b \times a

same vector hain."

Area symmetric hai lekin vector nahi — dono results opposite directions mein point karte hain; equal magnitude, flipped sign.

∣ a \times b ∣ = ∣ a ∣∣ b ∣ cos θ

Ye dot product ki magnitude hai — cross mein

sin θ

use hota hai (maximum jab perpendicular), dot mein

cos θ

(maximum jab parallel). "coS saath-saath, Sin sideways."

"Triangle ki area nikalne ke liye, main

∣ P Q \times P R ∣

leta hoon."

Ye parallelogram ki area deta hai; triangle uska aadha hai, isliye

\frac{1}{2}

se multiply karna zaroori hai.

"Right-hand rule ek alag fact hai jo formula ke upar add karna padta hai."

Component formula ka sign pattern pehle se hi ek perpendicular direction encode karta hai; right-hand rule sirf usi algebraic choice ki geometric picture hai.

a \times b = 0

matlab vectors perpendicular hain."

Ulta hai — perpendicular vectors maximum magnitude dete hain; zero magnitude matlab vectors parallel hain (ya ek zero vector hai).

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

Wrong sign — cyclic order

i \to j \to k

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

milta hai, isliye reverse karne par

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

Why questions

Dono inputs ke perpendicular hone ki demand se ek ki jagah do equations kyun milte hain?

Har perpendicularity condition,

c \cdot a = 0

aur

c \cdot b = 0

, ek linear equation hai; do vectors ke saath do milte hain, teen unknowns mein solutions ki poori line bachti hai (dekho Orthogonality).

Cross product ek vector kyun hai jabki dot product ek scalar hai?

Dono alag sawaalon ke jawaab dete hain: dot poochhta hai "ye kitna same direction mein point karte hain?" (ek number), cross poochhta hai "perpendicular direction kaun si hai aur kitna area?" — direction express karne ke liye vector chahiye.

Magnitude mein

cos θ

ki jagah

sin θ

kyun aata hai?

Base

a

ke upar parallelogram ki height

∣ b ∣ sin θ

hai; base times height se area

∣ a ∣∣ b ∣ sin θ

milta hai, isliye

b

ka perpendicular component matter karta hai.

2D mein genuine vector cross product define kyun nahi kar sakte?

2D mein "plane ke perpendicular" direction plane se bahar jaati hai, isliye koi single 2D vector do planar vectors ke perpendicular nahi ho sakta — result hold karne ke liye teesra dimension chahiye.

Formula automatically perpendicular vector kyun produce karta hai bina hum baad mein impose kiye?

Kyunki humne ise do perpendicularity equations se derive kiya tha; result ko wapas substitute karne par har term cancel ho jaata hai, isliye orthogonality built-in hai, baad mein check nahi ki jaati.

Scalar triple product

a \cdot (b \times c)

volume kyun deta hai?

b \times c

base parallelogram ka area ek perpendicular vector ke roop mein deta hai;

a

ke saath dot karne par

a

ko us height direction par project kiya jaata hai, isliye area × height = volume (dekho Scalar Triple Product aur Area and Volume).

Torque

r \cdot F

ki jagah

r \times F

kyun likha jaata hai?

Torque ko rotation axis aur turning strength chahiye — ek direction aur ek magnitude — jo sirf vector cross product provide kar sakta hai; ek scalar nahi bata sakta ki wo kis taraf spin karta hai (dekho Torque and Angular Momentum).

Top row mein

\hat{i}, \hat{j}, \hat{k}

wala determinant mnemonic actually kyun kaam karta hai?

3 \times 3

determinant ko top row ke along expand karne se teeno components exactly reproduce hote hain, signs included, isliye ye notation usi algebra ke liye ek memory aid hai.

Edge cases

a \times b

kya hota hai jab

θ = 0

ho (vectors parallel, same direction)?

Zero vector —

sin 0 = 0

matlab koi parallelogram nahi, isliye zero area aur koi defined perpendicular direction nahi.

a \times b

kya hota hai jab

θ = 18 0^{\circ}

ho (antiparallel)?

Phir bhi zero vector —

sin 18 0^{\circ} = 0

; dono vectors ek hi line par hain, koi area span nahi karte.

Agar ek input zero vector ho toh

a \times b

ka kya hota hai?

Result

0

hota hai; formula ke har term mein us vector ka component hota hai, aur geometrically zero-length side koi parallelogram span nahi karta.

Kis angle par

∣ a \times b ∣

maximum hota hai, aur woh maximum kya hai?

θ = 9 0^{\circ}

par, jahan

sin θ = 1

, maximum value

∣ a ∣∣ b ∣

milti hai — "most perpendicular" configuration.

Agar ek input scale karein,

a \times (k b)

, toh result ka kya hota hai?

Ye same factor se scale hota hai:

a \times (k b) = k (a \times b)

; agar

k < 0

ho toh direction bhi flip hoti hai kyunki area aur orientation dono sign ke saath respond karte hain.

Kya

a \times b

badalta hai agar

b

ki tail slide karein (translate karein) bina rotate kiye?

Nahi — cross product sirf vectors ki directions aur lengths par depend karta hai, un par nahi ki woh kahan placed hain; vectors position se free hote hain.

Connections

Dot Product — partner operation jisse ye traps constantly contrast karti hain
Determinants — sign pattern ka source jo log galat karte hain
Scalar Triple Product — jahan "area vector" "volume" ban jaata hai
Area and Volume — parallelogram/triangle factor-of-two trap
Orthogonality — kyun do perpendicularity conditions solutions ki poori line dete hain
Torque and Angular Momentum — kyun physics ko scalar nahi, vector chahiye