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

4.5.3 · Maths › Linear Algebra (Full)

Cross product KYA hai?

Ye sirf $R^{3}$ mein hi exist karta hai (cross product as a vector 3 dimensions ke liye special hai).

Perpendicular KYU aur area KYU? (scratch se derive karo)

Hum ek aisi recipe chahte hain jo do vectors ko ek teesre mein convert kare jo upar ke teen properties satisfy kare. Aao hum coordinate formula derive karte hain instead of yaad karne ke.

Step 1 — Perpendicularity demand karo. Likho $c = a \times b = (c_{1}, c_{2}, c_{3})$ . Hum require karte hain $c \cdot a = 0$ aur $c \cdot b = 0$ : $a_{1} c_{1} + a_{2} c_{2} + a_{3} c_{3} = 0, b_{1} c_{1} + b_{2} c_{2} + b_{3} c_{3} = 0.$ Ye step kyun? Property (1) kehta hai $c$ dono inputs ke orthogonal hai — ye teen unknowns $c_{1}, c_{2}, c_{3}$ mein do linear equations hain. Do equations, teen unknowns → solutions ki puri line, sab ek hi direction mein point karti hain.

Step 2 — System solve karo. Variables eliminate karne ki tarah treat karo. Do equations se solution (ek scalar tak) hai: $c = λ (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}) .$ Ye step kyun? Ise wapas plug karo: $a \cdot c = a_{1} (a_{2} b_{3} - a_{3} b_{2}) + a_{2} (a_{3} b_{1} - a_{1} b_{3}) + a_{3} (a_{1} b_{2} - a_{2} b_{1}) = 0$ — har term cancel ho jaata hai. To aisa koi bhi $c$ automatically perpendicular hai. Hume sirf scale $λ$ fix karna hai.

Step 3 — Area condition use karke scale fix karo. $λ = 1$ choose karo. Ab hum prove karte hain ki ye choice magnitude $∣ a ∣∣ b ∣ sin θ$ deta hai. Algebraic identity use karo (Lagrange's identity): $∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2} .$ Kyunki $a \cdot b = ∣ a ∣∣ b ∣ cos θ$ , $∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} (1 - cos^{2} θ) = ∣ a ∣^{2} ∣ b ∣^{2} sin^{2} θ .$ $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$ Ye step kyun? Parallelogram ka base $∣ a ∣$ hai aur uski height $∣ b ∣ sin θ$ hai ( $b$ ki tip se perpendicular giraao). Base × height = area = $∣ a ∣∣ b ∣ sin θ$ . To $λ = 1$ exactly sahi scale hai.

Figure — Cross product — formula, geometric meaning (area), right-hand rule

Right-hand rule (direction)

Apne right hand ki ungliyan $a$ ki taraf point karo, unhe $b$ ki taraf curl karo (angle $θ$ se); tumhara thumb $a \times b$ ke along point karega.

Consequences:

$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$ (cyclic order $i \to j \to k \to i$ ).
Anticommutative: $b \times a = - (a \times b)$ — hand direction swap karo, thumb flip ho jaata hai.
$a \times a = 0$ (parallel vectors → $sin 0 = 0$ → zero area).

Worked examples

Forecast-then-Verify

Recall Compute karne se pehle predict karo:

$u = (2, 0, 0)$ , $v = (0, 3, 0)$ . $u \times v$ kya hai aur uski length kya hai? Forecast: $x y$ -plane ke perpendicular → $\pm \hat{k}$ ke along; right-hand rule $+ \hat{k}$ deta hai; $2 \times 3$ rectangle ka area $= 6$ . Verify: $(0 \cdot 0 - 0 \cdot 3, 0 \cdot 0 - 2 \cdot 0, 2 \cdot 3 - 0 \cdot 0) = (0, 0, 6)$ . Length $6$ . ✓ Bilkul forecast ke jaisa.

Common mistakes (steel-manned)

80/20 — wo 20% jo 80% deta hai

Output ek vector ⊥ to both inputs hai.
Length = area $= ∣ a ∣∣ b ∣ sin θ$ ; triangle area aadhi hoti hai.
Right-hand rule direction deta hai; order swap karne se sign flip ho jaata hai.
$3 \times 3$ determinant se compute karo (middle sign dhyan se dekhna).

Recall Feynman: 12-saal ke bachche ko samjhao

Socho do lathiyaan ek end se chipki hain, pizza ke slice ki tarah khuli hain. Unke beech ek rubber sheet kheencho — ye ek tilted flap hai. Cross product ek naya arrow hai jo us flap se bilkul seedha upar khada hai, perfectly upright. Wo arrow jitna lamba hoga, flap utni hi badi hogi (zyada area). Agar do lathiyaan ek hi direction mein point karti hain to koi flap nahi hoga, isliye arrow ki length zero hogi. "Upar" kaun si taraf hai? Apna right hand use karo: ungliyan pehli laathi se doosri ki taraf sweep karo, aur tumhara thumb upar dikhayega.

Connections

Dot Product — partner operation; $∣ a \times b ∣^{2} + (a \cdot b)^{2} = ∣ a ∣^{2} ∣ b ∣^{2}$
Determinants — formula hai hi ek $3 \times 3$ determinant; signs cofactors se aate hain
Scalar Triple Product — $a \cdot (b \times c)$ = parallelepiped ka signed volume
Area and Volume — parallelogram/triangle area, oriented area
Orthogonality — perpendicularity do linear equations kyun deta hai
Torque and Angular Momentum — $r \times F$ ke physics applications

Flashcards

Cross product kya output karta hai (type)?

Ek vector (in

R^{3}

), dono inputs ke perpendicular.

a \times b

ka magnitude?

∣ a ∣∣ b ∣ sin θ

= unse span hone wale parallelogram ka area.

∣ a \times b ∣

ka geometric meaning?

a

aur

b

se span hone wale parallelogram ka area.

Do edge vectors

u, v

se triangle area?

\frac{1}{2} ∣ u \times v ∣

a \times b

ka component formula?

(a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})

Kya cross product commutative hai?

Nahi, anticommutative:

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

a \times a

kya hai?

0

(parallel vectors,

sin 0 = 0

Right-hand rule statement?

Ungliyan

a

ke along,

b

ki taraf curl karo; thumb

a \times b

deta hai.

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

\hat{k}

(cyclic:

i \to j \to k \to i

Dot aur cross ko link karne wali Lagrange's identity?

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

Dot kaun sa trig function use karta hai cross ke comparison mein?

Dot

cos θ

use karta hai (max parallel); cross magnitude

sin θ

use karta hai (max perpendicular).

a \times b

a

ke perpendicular kyun hai?

a \cdot (a \times b) = 0

algebraically; do orthogonality equations se bana hai.