4.5.3 · D1Cross product — formula, geometric meaning (area), right-han
3D space mein do arrows, apni tails se jude hue, ek flat tilted flap banate hain. Cross product ek teesra arrow banata hai jo us flap se seedha upar khada hota hai, aur jis ki length us flap ka area ke barabar hoti hai. Ye sentence samajhne ke liye tumhe sirf itna jaanna chahiye: ek arrow (vector) kya hota hai, "length" aur "angle" ka matlab kya hai, aur ek right hand "upar" aur "neeche" mein farq kaise karta hai.
Is page pe koi assumption nahi hai. Parent note Cross product — formula, geometric meaning (area), right-hand rule ko touch karne se pehle, ensure karo ki neeche har symbol tumhare dimaag mein ek picture hai, na ki page pe ek squiggle.
R 3 R real numbers ka set hai (ek endless number line pe har point). R 3 ( x , y , z ) right , ek kitna upar , ek kitna tumhari taraf . Ek room ka corner imagine karo: floor ki edges aur vertical edge teen directions hain.
Topic ko iske zaruurat kyun hai: cross product wohi ek operation hai jo sirf yahin rehta hai. Flat 2D mein koi "page se upar" direction nahi hoti jo khud ek 2D arrow ho, isliye yeh trick kaam nahi karti. Teen axes perpendicular arrow ko khadha hone ki jagah dete hain.
i ^ , j ^ , k ^ basis hain — standard rulers jinse hum sab kuch measure karte hain (§4 mein define kiya gaya hai).
Ek vector ek arrow hai: iske paas ek length hai (kitna lamba) aur ek direction (kis taraf point karta hai). Hum ise a a = ( a 1 , a 2 , a 3 ) a 1 a 2 a 3
Intuition Number vs. arrowEk scalar ek akela number hai (5 − 2.3 vector ek direction bhi carry karta hai. Dot product ek scalar dega; cross product ek vector dega. Ye jaanna ki kaunsa type output aata hai, topic ka aadha samajhna hai.
Subscripts a 1 , a 2 , a 3 a 2 a
Definition Magnitude (length)∣ a ∣ a arrow ki length hai. Room mein Pythagoras ko do baar stack karke,
∣ a ∣ = a 1 2 + a 2 2 + a 3 2 .
Square root kyun? Chhote bars puri arrow ko ek honest length mein badal dete hain. Pythagoras kehta hai ek box ka diagonal squared edges ke sum ka square root hota hai — to yeh formula bas "box ka diagonal jiske sides a 1 , a 2 , a 3
Worked example Number feel karoa = ( 2 , 0 , 0 ) ∣ a ∣ = 4 = 2
Parent ka area formula ∣ a ∣ ∣ b ∣ ∣ a × b ∣
θ a b angle , jise θ 0 ∘ 18 0 ∘
Ab us angle ke do special ratios har jagah aate hain:
Definition Ek right triangle pecos θ sin θ b a b hypotenuse ) hai.
cos θ = hypotenuse adjacent b along- part (a sin θ = hypotenuse opposite b sideways part (a
sin cos height (perpendicular gap) exactly ∣ b ∣ sin θ sideways part. Area = base × height = ∣ a ∣ ⋅ ∣ b ∣ sin θ sin align na hona (sideways-ness) measure karta hai. Dot product cos alignment measure karta hai. Yahi hai poora "coS together, Sin sideways" mnemonic.
Extremes ki sanity:
θ = 0 ∘ sin 0 = 0 zero area . Flap flat hai, koi flap hi nahi.θ = 9 0 ∘ sin 9 0 ∘ = 1 θ = 18 0 ∘ sin 18 0 ∘ = 0
Definition Unit basis vectorsEk unit vector exactly 1 ^ i ^ = ( 1 , 0 , 0 ) , j ^ = ( 0 , 1 , 0 ) , k ^ = ( 0 , 0 , 1 ) .
Topic ko iske zaruurat kyun hai: determinant mnemonic
a × b = i ^ a 1 b 1 j ^ a 2 b 2 k ^ a 3 b 3 i ^ , j ^ , k ^
Definition Dot product (partner operation)Dot product matching parts multiply karke add karta hai:
a ⋅ b = a 1 b 1 + a 2 b 2 + a 3 b 3 = ∣ a ∣∣ b ∣ cos θ . number (scalar). Yeh bada hota hai jab arrows same direction mein point karte hain, aur zero hota hai jab perpendicular hain .
Topic ko iske zaruurat kyun hai, do jagah:
Perpendicularity test. "c a c ⋅ a = 0 do dot-product equations zero ke barabar set hai.Lagrange's identity. Clean link ∣ a × b ∣ 2 = ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b ) 2
Full builds in Dot Product aur Orthogonality .
2 × 2 p r q s = p s − q r ( p , q ) ( r , s ) signed area hai: positive agar counter-clockwise jaate hain, negative agar clockwise.
Definition Top row ke along3 × 3 i ^ a 1 b 1 j ^ a 2 b 2 k ^ a 3 b 3 = i ^ a 2 b 2 a 3 b 3 − j ^ a 1 b 1 a 3 b 3 + k ^ a 1 b 1 a 2 b 2 . plus i ^ j ^ k ^ . Beech mein woh akela minus wohi jagah hai jahan almost har sign error paida hoti hai.
Ise expand karo aur tumhe exactly parent ka component formula milega
a × b = ( a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 ) . a 3 b 1 − a 1 b 3 Determinants .
Intuition Sirf algebra kaafi nahi kyunEk flat flap ke do perpendicular directions hoti hain: seedha upar aur seedha neeche. Formula quietly ek choose karta hai (apne signs ke zariye); right-hand rule us choice ki physical picture hai.
Apni right fingers a θ b thumb a × b b × a = − ( a × b )
Tumhe yeh jaanna chahiye taaki pata chale ki topic ka matlab kaun sa do upright arrows mein se hai. Yeh physics mein har jagah reappear karta hai — dekho Torque and Angular Momentum .
Real numbers and R3 space
Length bars sqrt of squares
Angle theta between arrows
sin theta gives sideways height
cos theta gives alignment
Dot product alignment number
Perpendicular test dot equals zero
Left side ka har box solid hona chahiye tab hi right side ka "CROSS PRODUCT" box sense banata hai.
Right side cover karo aur reveal karne se pehle har ek ka jawab dene ki koshish karo.
R 3 Har triple ( x , y , z )
Ek vector kya do cheezein carry karta hai? Ek length aur ek direction (yeh ek arrow hai).
∣ a ∣ a 1 2 + a 2 2 + a 3 2 ; yeh Pythagoras hai — box ka diagonal jiske edges
a 1 , a 2 , a 3 hain.
Tail-to-tail triangle pe, kaun sa ratio sin θ opposite / hypotenuse — arrow ka sideways part.
Area sin θ cos θ Parallelogram ki height
∣ b ∣ sin θ hai (perpendicular gap); base × height ko sideways part chahiye.
Jab do arrows parallel hoon to sin θ 0
i ^ , j ^ , k ^ ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 )
"c a p r q s p s − q r
Top-row expansion mein kaun sa sign alag hai, aur yeh kyun matter karta hai? Beech ka j ^ minus hai; isse bhool jaana poori second component ko flip kar deta hai.
Kaun sa haath, aur fingers vs. thumb kya karte hain? Right hand; fingers
a → b sweep karte hain, thumb
a × b ke along point karta hai.
Dot Product — alignment number aur perpendicular test jo yahin build huaOrthogonality — kyun "perpendicular" ek equation ban jaata hai jo zero ke barabar set hoti haiDeterminants — §6 ka sign machine poori tarah seArea and Volume — jahan sin θ Scalar Triple Product — agla rung: cross ka dot volume deta haiTorque and Angular Momentum — right-hand rule physics mein