4.5.1 · D1Vectors in ℝⁿ — operations, geometric interpretation
Ek vector ek number ki list hai jo saath-saath ek geometric arrow bhi hai — aur jab ek baar un arrows ko add aur stretch kar sako, toh plain arithmetic geometry ban jaati hai . Parent page ki har cheez (length, angle, perpendicularity) usi ek dual meaning se nikalti hai: number-list ↔ arrow.
Is page mein kuch bhi assumed nahi hai . Parent note mein jo bhi squiggle, letter, aur picture use hua, wo sab yahan se ground up pe banaya jayega, ek aisi order mein jahan har brick apne pehle wali brick pe tiki ho.
Arrows se pehle, wo zameen chahiye jis par wo rehte hain.
Definition Real number aur symbolR real number wo koi bhi point hai jo tum ek endless ruler par mark kar sako: whole numbers, fractions, negatives, 2 π R "real numbers ke poore set" ka shorthand hai — ek saath poora ruler.
R 0 v 1 ∈ R v 1 koi point hai ." Chota "∈ "is an element of / ke andar rehta hai."
Hume R
Ek ruler tumhe ek line par locate karta hai. Map par apne aap ko locate karne ke liye do rulers chahiye (east–west, north–south). Room mein ek point locate karne ke liye teen chahiye (add up–down).
R n R n n n kitni independent directions hain.
R 1 R 2 R 3 R n n
R n R n Yeh kyun sahi lagta hai: raised number dikhta hai jaise exponent ho. Fix yeh hai: yahan superscript ek counter of axes hai, multiplication nahi. R 3 R × R × R hai , lekin kuch cube nahi ho raha.)
Definition Vector, component, aur subscripti vector v R n bold print karte hain (v v = ( v 1 , v 2 , … , v n ) .
Har v i component hai: ruler number i
Chota subscript i pointer/address hai: "v 2 … n
Intuition Ek list ke liye do pictures — poora jaaduExactly wahi list ( 3 , 4 )
ek point — ek address: "east-3, north-4 par dot";
ek arrow — ek journey: "origin se shuru karo, 3 east phir 4 north chalo, aur jahan utro wahan tak seedha arrow khicho."
Arrow picture woh hai jo hume add aur stretch karne deti hai; point picture woh hai jo hume locate karne deti hai. Dono ko ek saath mind mein rakho.
0 0 = ( 0 , 0 , … , 0 ) origin hai — "you are here / start" point jahan saare arrows rooted hain. Bold zero = zero vector (zeros ki poori list), plain number 0
Column notation. Parent vectors ko vertically stack karke likhta hai:
v = v 1 ⋮ v n . wahi list hai , bas left-to-right ki jagah top-to-bottom likhi gayi. Tall square brackets ek container hain; vertical dots ⋮ ( v 1 , … , v n )
Parent jo kuch bhi banata hai woh exactly do moves par chalta hai.
Ek scalar sirf ek ordinary single real number hai (R scale karne ke liye use hota hai — stretch ya shrink karna. Hum ise "scalar" kehte hain (na ki "vector") exactly yeh flag karne ke liye: "yeh ek number arrow nahi hai, yeh ek stretch-factor hai." Hum scalars ko usually plain letters jaise c
u + v c v u + v = ( u 1 + v 1 , … , u n + v n ) , c v = ( c v 1 , … , c v n ) . component-wise kaam karte hain: operation ek ruler pe ek baar karo.
Intuition Component-wise kyun, aur har ek kaisa dikhta hai3 units east move karna kabhi nahi badalta kitna north ho — rulers independent hain. Toh honest rule sirf yahi hai ki matching entries ko combine karo aur baaki ko chhod do.
Addition = tip-to-tail: v u u + v c ∣ c ∣ c < 0 flip bhi karta hai opposite direction mein point karne ke liye. Agar c = 0 0
Definition Absolute value bars∣ c ∣ ∣ c ∣ c ∣ − 3 ∣ = 3 ∣3∣ = 3 0 c length kabhi negative nahi ho sakti, chahe stretch-factor ho.
"Yeh arrow kitna lamba hai?" — yeh poochne ke liye parent Pythagoras invoke karta hai. Do symbols aate hain.
v i 2
v i 2 v i × v i area jiska side v i khatam karta hai — ( − 4 ) 2 = 16 x x undo karta hai. 25 = 5 5 2 = 25
∥ v ∥ ∥ v ∥ v v single non-negative number hai: arrow ki straight-line length. Double bars (single nahi) signal dete hain "poore vector ki length," plain number ke liye ∣ c ∣ ∥ v ∥ = v 1 2 + v 2 2 + ⋯ + v n 2 .
Intuition Exactly yahi formula kyun (aur square-root sahi tool kyun hai)Plane mein, arrow ( v 1 , v 2 ) hypotenuse hai jiske legs v 1 v 2 leg 2 + leg 2 = hyp 2 squared length v 1 2 + v 2 2 ∥ 0 ∥ = 0 + ⋯ + 0 = 0 0
Definition Summation sign∑ i = 1 ∑ n v i 2 i 1 , 2 , … , n v i 2 ∑ loop instruction hai: neeche "i = 1 n ∥ v ∥ = ∑ i = 1 n v i 2
Definition Unit vector aur hatv ^ unit vector ki length exactly 1 direction bina kisi size ke. Kisi bhi nonzero v normalize karo: har component ko length se divide karo,
v ^ = ∥ v ∥ v . ^ ∥ v ∥ same factor se shrink hota hai — toh direction bachti hai jabki length 1
Parent ka climax arithmetic ko angles mein turn karta hai. Chaar naye symbols.
⋅ u ⋅ v = u 1 v 1 + u 2 v 2 + ⋯ + u n v n . Matching components multiply karo, phir sab ko ek plain number (ek scalar) mein add karo. Raised dot "⋅ do vectors ke beech jo vector nahi balki ek single alignment score return karta hai.
θ cos
θ angle name karta hai jo ek tail share karte do arrows ke beech hota hai — ek se doosre tak turn ki matra.cos θ dial reading − 1 + 1 hai jo batata hai do directions kitna aligned hain: cos 0 ∘ = 1 cos 9 0 ∘ = 0 cos 18 0 ∘ = − 1
Definition Perpendicular symbol⊥ ⟺
u ⊥ v u v perpendicular (right angle par) hai."⟺ if and only if " — ek do-taraf ka raasta: left tab sach hai exactly jab right sach ho.
u ⊥ v ⟺ u ⋅ v = 0. Zero kyun? Right angle θ = 9 0 ∘ cos 9 0 ∘ = 0 0 detect karne ke liye flip karta hai.
Recall Quick self-check: kaunsa symbol kya return karta hai?Kya ∥ v ∥ u ⋅ v c v v ⋅ v ∥ v ∥ 2
Vector v = list of components
Point picture and Arrow picture
Scalar multiply stretch and flip
Linear Combinations and Span
Norm length via Pythagoras
Unit vectors and normalize
Perpendicular test dot equals 0
Yeh map parent note ka miniature hai: real numbers R n Linear Combinations and Span , Dot Product and Orthogonality , aur Norms and Distance in Rn mein hain, aur poori structure Vector Spaces — Axioms mein formalize ki gayi hai.
Self-test: kya tum har ek ko ek plain sentence mein keh sakte ho reveal karne se pehle ?
R Saare real numbers ka set — ek endless ruler par har point.
R n n n
∈ "is an element of / ke andar rehta hai," e.g. v i ∈ R
v Ek vector — R n
v i i i i
0 Origin / zero vector, saare components zero — jahan arrows rooted hain.
scalar c Ek single real number jo vector ko stretch/shrink karne ke liye use hota hai.
component-wise Operation ek matching entry at a time karo.
c v Scaling: length ko ∣ c ∣ c < 0 c = 0 0
∣ c ∣ Absolute value — 0 c
v i 2 Squaring (side-v i
∥ v ∥ Norm — arrow ki straight-line length, ek non-negative number.
∑ i = 1 n Summation loop: terms add karo jaise i 1 n
v ^ Unit vector — v 1 v /∥ v ∥
u ⋅ v Dot product — matching components ke products ka sum; ek single scalar.
θ Do arrows ke beech angle jo ek tail share karte hain.
cos θ Alignment dial − 1 + 1 1 0 − 1
⊥ Perpendicular (right angle) — tab hold karta hai exactly jab dot product 0
⟺ "If and only if" — dono sides saath sach hoti hain, ek do-taraf implication.
