1.1.9
Ek akela arrow dobara banaya ja sakta hai do aur arrows se jo chosen directions mein rakhe gaye hon. Kisi vector ko resolve karna matlab hai yeh poochna: "is vector ka kitna hissa THIS axis ki taraf point karta hai, aur kitna THAT axis ki taraf?" Jab woh do numbers pata chal jaayein, toh original vector poori tarah describe ho jaata hai — humne "length + angle" wali picture ko ek "do-number" wali picture se replace kar diya jo add, subtract, aur calculate karne mein bahut zyada aasaan hai.
Intuition Kyun takleef uthayein?Vectors ko graphically add karna (tip-to-tail) tedious aur inaccurate hota hai. Lekin ek fixed axis ke saath numbers ordinary numbers ki tarah add ho jaate hain . Toh trick yeh hai: har vector ko same set of axes ke saath pieces mein tod do, pieces ko alag se add karo, phir reassemble karo. Resolution geometry ko arithmetic mein badal deta hai.
Definition Rectangular componentsA θ components perpendicular axes par projections hain:
A x = A cos θ , A y = A sin θ A = ∣ A ∣ A = A x x ^ + A y y ^
A right triangle banta hai jiska hypotenuse A
Woh side jo θ adjacent hai woh x ke saath hai. Cosine ki definition se: cos θ = hyp adjacent = A A x A x = A cos θ (Kyun? cosine literally hypotenuse ki shadow ko adjacent direction mein measure karta hai.)
Woh side jo θ opposite hai woh y ke saath hai: sin θ = A A y A y = A sin θ
Magnitude aur angle recover karna (inverse problem): usi triangle par Pythagoras deta hai
A = A x 2 + A y 2 , tan θ = A x A y .
Ek component bas ek shadow (projection) hai. Yeh pata karne ke liye ki A kisi bhi direction n ^ n ^
DOT PRODUCT kyun? Isse derive karo: agar n ^ α A θ ϕ = θ − α A cos ( θ − α ) = A cos θ cos α + A sin θ sin α = A x cos α + A y sin α . n ^ = ( cos α , sin α ) A ⋅ n ^
Definition Oblique resolutionAgar tum do aise axes par insist karo jo not 9 0 ∘ wapas A sakein. Directions u ^ , v ^ A = a u ^ + b v ^ a u ^ + b v ^ parallelogram components hain, generally perpendicular projections ke =
Common mistake Steel-man: "Oblique components ke liye bhi basA ⋅ u ^ Kyun sahi lagta hai: projection ne x–y axes ke liye hamesha answer diya, toh surely har jagah kaam karega.
Kyun galat hai: x–y ke liye projection parallelogram component ke equal hota hai sirf isliye kyunki axes perpendicular hain . Oblique axes ke liye u ^ v ^ Fix: a u ^ + b v ^ = A
Worked example Example 2 — inclined plane (tilted axes!)W = m g θ W incline ke saath aur perpendicular resolve karo.
Axis choose karo incline ke saath (down-slope) aur perpendicular (surface mein).
Component along incline: W ∥ = W sin θ sin θ
Component perpendicular: W ⊥ = W cos θ N
Kyun humne axes tilt kiye: motion slope ke saath hoti hai, toh ek tilted axis ek component ko driving force aur doosre ko balanced force bana deta hai — x–y se bahut zyada simple.
Worked example Example 3 — projection onto an arbitrary directionA = 3 x ^ + 4 y ^ n ^ α = 5 3 ∘ n ^ ≈ 0.6 x ^ + 0.8 y ^
A n ^ = A ⋅ n ^ = 3 ( 0.6 ) + 4 ( 0.8 ) = 1.8 + 3.2 = 5.0 Kyun? dot product = projection. Sanity check: ∣ A ∣ = 5 n ^ A 5 3 ∘ tan − 1 ( 4/3 ) 5
Recall Feynman style: ek 12-saal ke bachche ko samjhaoSocho ek kite ki string ek tiraafi angle par kheench rahi hai. Us khinchaav ka ek part tumhe sideways kheenchta hai aur ek part tumhe upar . Vector ko resolve karna bas iska matlab hai "kitna sideways?" aur "kitna upar?" alag alag naapna. Tiraafi khinchaav wahin hai jahan woh do alag khinchaav ek saath hote hain. Agar tum khinchaav ko kisi aur direction mein jaanna chahte ho — maan lo kisi ped ki taraf — toh apna "shadow ruler" ped ki taraf tilt karo aur shadow kitni lambi hai woh padho. Woh shadow component hai.
Mnemonic Yaad karo kis ko cos milta hai"COS hugs the Angle." Woh side jo angle ko touch karti hai (adjacent) woh cos use karti hai; woh side jo aar paar hai (opposite) woh sin use karti hai. Slogan: Adjacent–Cos, Opposite–Sin .
A x = A cos θ θ x-axis se measure kiya jaaye?Oblique axes ke liye projection component ke same kyun nahi hoti?
Kaun sa ek operation kisi bhi unit vector ke saath component deta hai?
Rectangular components of A θ A x = A cos θ , A y = A sin θ
Adjacent side cosine kyun use karta hai? cos θ = adjacent / hyp = A cos θ
Components se magnitude recover karo Components se direction recover karo θ = tan − 1 ( A y / A x )
Component of A n ^ A ⋅ n ^ = A cos ϕ (the projection)
Dot product projection kyun hai? A cos ( θ − α ) = A x cos α + A y sin α = A ⋅ n ^ expand karke
θ W sin θ
θ W cos θ
Projection parallelogram component ke equal kab hoti hai? Sirf tab jab axes mutually perpendicular hon
Vectors ko resolve kyun karte hain? Fixed axis ke saath components ordinary numbers ki tarah add hote hain, geometry ko arithmetic mein badal dete hain
Graphical vector addition clumsy
Magnitude and angle recovered
Projection onto any axis n
Expand cos theta minus alpha
Simultaneous equations for a and b
