1.1.8
Ek vector ek "journey" hai jisme ek length aur ek direction hoti hai. Do vectors ko add karna matlab hai: pehli journey karo, phir doosri — aakhir mein kahan pohonche? Start se finish tak ka seedha arrow resultant hai. Neeche sab kuch (triangle law, parallelogram law, magnitude formula) usi ek idea ki bookkeeping hai.
Definition Vector additionDo vectors A B sum (ya resultant ) R = A + B A B
"Journey" idea se seedhe do key facts milte hain:
Addition commutative hoti hai: A + B = B + A
Sirf same quantities hi add ho sakti hain (force + force, velocity + velocity) — force ko velocity se kabhi nahi.
Definition Vector addition ka Triangle lawAgar do vectors ko ek triangle ki do sides se magnitude aur direction mein represent kiya jaye jo same order (head-to-tail) mein li gayi hain, toh resultant ko teesri side se opposite order mein represent kiya jaata hai (pehle ka tail se doosre ka head).
WHY head-to-tail? Kyunki B jahaan pehli journey khatam hui thi . B
Definition Parallelogram lawAgar ek point par act karne wale do vectors ko ek parallelogram ki do adjacent sides se represent kiya jaye (jo same point se, tail-to-tail draw ki gayi hain), toh resultant ko us point se guzarne wale parallelogram ke diagonal se represent kiya jaata hai.
WHY yeh same cheez hai: Parallelogram mein, opposite side B A B Parallelogram law = triangle law alag costume mein.
Maano A B beech ka angle (tail-to-tail) θ A
Step 1 — Coordinates set karo.
Common tail O A
A ( A , 0 ) B A θ ( B cos θ , B sin θ )
Yeh step kyun? Components ki wajah se hum x aur y independently add kar sakte hain, geometry ko arithmetic mein badal dete hain.
Step 2 — B A (head-to-tail). R R x = A + B cos θ , R y = B sin θ .
Step 3 — Pythagoras (head, O R x R y R = R x 2 + R y 2 = ( A + B cos θ ) 2 + ( B sin θ ) 2 .
Step 4 — Expand aur simplify karo.
R 2 = A 2 + 2 A B cos θ + B 2 cos 2 θ + B 2 sin 2 θ . cos 2 θ + sin 2 θ = 1
Step 5 — Direction. Resultant A α tan α = R x R y = A + B cos θ B sin θ
Yeh step kyun? B tan α = adjacent opposite
Intuition Sanity checks (Forecast-then-Verify)
θ = 0 ∘ R = A 2 + B 2 + 2 A B = A + B θ = 18 0 ∘ R = A 2 + B 2 − 2 A B = ∣ A − B ∣ θ = 9 0 ∘ R = A 2 + B 2 [ ∣ A − B ∣ , A + B ]
Common mistake "Bas magnitudes add karo:3 + 4 = 7 Kyun sahi lagta hai: Ordinary numbers ke liye, addition sirf values jodna hoti hai. Galti: vectors direction carry karte hain; sirf tab R = A + B θ = 0 Fix: hamesha R = A 2 + B 2 + 2 A B cos θ
Common mistake "Triangle law mein vectors ko tip-to-tip add karo."Kyun sahi lagta hai: Unhe same point se draw karna neat lagta hai. Galti: triangle law ko head-to-tail chahiye; tip-to-tip sum ki jagah difference deta hai. Fix: A + B B A parallelogram law ke liye hai, jahan diagonal — koi side nahi — sum hoti hai.)
tan α Kyun sahi lagta hai: angles toh angles hote hain. Galti: har formula mein θ do vectors ke beech ka angle hai, tail-to-tail measure kiya hua. Fix: dono vectors ko ek common origin se redraw karo; unke beech ka wedge θ
Triangle law statement Do vectors ek triangle ki do sides ke roop mein same order mein (head-to-tail); resultant teesri side hai opposite order mein.
Parallelogram law statement Do vectors ek common point se adjacent sides ke roop mein; resultant us point se guzarne wala diagonal hai.
Magnitude of resultant of A , B θ Direction of resultant (angle with A tan α = A + B cos θ B sin θ
Maximum possible resultant A + B θ = 0 ∘
Minimum possible resultant ∣ A − B ∣ θ = 18 0 ∘
Resultant when θ = 9 0 ∘ Formula mein θ Do vectors ke beech ka angle, tail-to-tail measure kiya hua.
Triangle & parallelogram laws equivalent kyun hain? Parallelogram ki opposite sides equal & parallel hoti hain, isliye diagonal ek aisa head-to-tail triangle close karta hai jisme same do vectors hain.
Do equal vectors ke liye resultant kidhar point karta hai? Angle bisector ke saath (symmetry se).
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho tum 3 steps east chalte ho, phir 4 steps north. Tum ghar se 7 steps dur nahi pohonchte — tum 5 steps dur pohonchte ho, ek diagonal par! "Arrows" (vectors) add karna matlab hai ek ke baad ek chalna aur poochna "main jahaan se chala tha, wahan se mera ghar kitna dur hai aur kis direction mein?" Triangle law unhi do walks ko triangle ki do sides ki tarah draw karta hai, aur ghar ka rasta teesri side hai. Parallelogram law usi trick ka ek tircha box use karke version hai.
"Head-to-Tail to find the trail; Tail-to-Tail, the Diagonal prevails."
Triangle = Head-to-Tail (side use karo). Parallelogram = Tail-to-Tail (Diagonal sum hai).
Aur magnitude ke liye: "A-squared, B-squared, plus two-A-B-cos."
Vectors — components and unit vectors (i,j method upar ke Step 1–2 ko generalize karta hai)Subtraction of vectors and the difference vector (A − B θ → 18 0 ∘ − θ Resolution of a vector into components Relative velocity (real-world vector addition: boat + river)Law of cosines (magnitude formula hi disguise mein cosine rule hai)Scalar (dot) product (woh cos θ
Vector addition = resultant
Triangle law head-to-tail
Parallelogram law tail-to-tail
R = sqrt of A2 + B2 + 2AB cos theta
