1.1.12
Dot product poochta hai "do vectors kitna SAME direction mein point karte hain?" — yeh ek number (scalar) return karta hai.
Cross product iska ulta poochta hai: "do vectors kitna PERPENDICULAR / door hain ek doosre se?" — aur sirf ek number ki jagah, yeh ek naya vector return karta hai jo un dono vectors ke plane se bahar point karta hai.
WHY ek vector? Kyunki rotation, torque aur area — teeno ko space mein ek direction chahiye (wheel kis taraf spin karta hai? surface kis taraf face karta hai?). Ek plain number yeh encode nahi kar sakta — lekin plane ke perpendicular ek vector kar sakta hai.
Do vectors A B θ vector hai jo is tarah define hota hai:
Magnitude: ∣ A × B ∣ = A B sin θ Direction: perpendicular to the plane jisme A B right-hand rule se.
sin θ
Agar A ∥ B θ = 0 sin 0 = 0 zero hai. Parallel vectors mein koi "spread" nahi hota.
Agar A ⊥ B θ = 9 0 ∘ sin 9 0 ∘ = 1 maximum = A B
Toh cross product exactly tab peak karta hai jab dot product zero ho, aur vice versa.
A B sin θ A B parallelogram banate hain.
A A height hai B A B sin θ = × = A ⋅ ( B sin θ ) = A B sin θ
∣ A × B ∣ = A B sin θ = area of the parallelogram
Intuition Right-hand rule ("curl" version)Apne right hand ki ungliyan A curl karo B θ thumb A × B
Yeh directly batata hai ki cross product anti-commutative hai:
A × B = − B × A Kyun? B A
Worked example Unit vectors ek cycle banate hainRight-handed axes ke saath:
i ^ × j ^ = k ^ , j ^ × k ^ = i ^ , k ^ × i ^ = j ^ i ^ × i ^ = 0 Cycle ke liye Mnemonic: i → j → k → i + −
Derivation Unit-vector rules se build karnaA = A x i ^ + A y j ^ + A z k ^ B = B x i ^ + B y j ^ + B z k ^ A × B i ^ × i ^ A × B = ( A y B z − A z B y ) i ^ + ( A z B x − A x B z ) j ^ + ( A x B y − A y B x ) k ^
Worked example Example 1 — Pure component computationA = 2 i ^ + 3 j ^ + k ^ B = i ^ − j ^ + 4 k ^ A × B
i ^ A y B z − A z B y = ( 3 ) ( 4 ) − ( 1 ) ( − 1 ) = 12 + 1 = 13 Yeh step kyun? i ^ j ^ − ( A x B z − A z B x ) = − [( 2 ) ( 4 ) − ( 1 ) ( 1 )] = − ( 8 − 1 ) = − 7 Minus kyun? Middle term determinant ka − k ^ A x B y − A y B x = ( 2 ) ( − 1 ) − ( 3 ) ( 1 ) = − 2 − 3 = − 5
A × B = 13 i ^ − 7 j ^ − 5 k ^
Worked example Example 2 — Triangle ka areaTriangle ke vertices hain P ( 0 , 0 , 0 ) Q ( 1 , 2 , 0 ) R ( 2 , 0 , 0 )
P P Q = ( 1 , 2 , 0 ) P R = ( 2 , 0 , 0 ) Same vertex se kyun? Cross product ke liye dono vectors ka tail common hona chahiye.
P Q × P R = i ^ 1 2 j ^ 2 0 k ^ 0 0 = ( 2 ⋅ 0 − 0 ⋅ 0 ) i ^ − ( 0 − 0 ) j ^ + ( 0 − 4 ) k ^ = − 4 k ^ = 2 1 ∣ − 4 k ^ ∣ = 2 1 ( 4 ) = 2 Check: base P R = 2 = 2 Q y = 2 1 ( 2 ) ( 2 ) = 2
Worked example Example 3 — TorqueEk force F = ( 0 , 5 , 0 ) N r = ( 2 , 0 , 0 ) m τ = r × F
τ = i ^ 2 0 j ^ 0 5 k ^ 0 0 = ( 0 ) i ^ − ( 0 ) j ^ + ( 2 ⋅ 5 − 0 ) k ^ = 10 k ^ N⋅m 10 + k ^ counter-clockwise rotation, right-hand rule se consistent. Yeh step kyun? r x F y i ^ × j ^ = k ^
τ = r × F , ∣ τ ∣ = r F sin θ F r perpendicular hai, turning produce karta hai. F sin θ r τ
Common mistake "Cross product ek number deta hai"Kyun aisa lagta hai: dot product scalar deta hai, toh habit se scalar expect hota hai.
Fix: cross product ek vector deta hai — uski ek direction hoti hai (rotation axis). ∣ A × B ∣ A × B
A × B = B × A Kyun aisa lagta hai: numbers ki multiplication commutative hoti hai.
Fix: cross product anti -commutative hai: A × B = − B × A
Common mistake Determinant mein middle sign bhool jaanaKyun aisa lagta hai: tum left-to-right expand karte ho sab plus signs ke saath.
Fix: cofactor signs hote hain + , − , + j ^
sin θ cos θ Kyun aisa lagta hai: dot product A B cos θ Fix: Dot cos sin
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho tum ek darwaza dhakka de rahe ho. Agar tum seedha hinge ki taraf dhakka do, kuch nahi hoga. Agar tum edge par, sideways dhakka do, toh woh aasaani se khul jaata hai. Cross product wahi math hai jo kehta hai: "sirf sideways dhakka count karta hai, aur darwaza apne hinge-line ke around ghoomta hai." Yeh tumhe dono cheezein batata hai — kitna zor se ghoomta hai (number A B sin θ spin-axis kis taraf hai (jab tum pehle arrow se doosre ki taraf ungliyan curl karo toh tumhari right thumb). Do arrows jo flat rakhe hain woh ek tedha patch banate hain — cross product yeh bhi measure karta hai ki woh patch kitna bada hai .
"Sin for Spin, Cos for Close." Cross→sin cos Cycle: i ^ → j ^ → k ^ → i ^ + − Right hand: ungliyan A ke saath, curl karo B ki taraf, thumb hai A×B .
A × B A B sin θ
A × B A , B ke plane ke perpendicular, right-hand rule se
Kya cross product commutative hai? Nahi;
A × B = − B × A (anti-commutative)
A × B = 0 Jab vectors parallel/antiparallel hon (θ = 0 18 0 ∘ sin θ = 0
∣ A × B ∣ A aur
B se bane parallelogram ka area
Do edge vectors se triangle ka area? i ^ × j ^ = ? k ^
k ^ × j ^ = ? − i ^
Torque ko cross product ki form mein formula kya hai? Determinant expansion mein j ^ Minus (+ , − , +
Cross vs dot — sin Cross product sin θ cos θ
A × B x A y B z − A z B y
Triangle area = half A x B
Determinant component formula
