Dot product — formula, cosine formula, Cauchy-Schwarz inequality proof: Foundations

4.5.2 · D1 · Maths › Linear Algebra (Full) › Dot product — formula, cosine formula, Cauchy-Schwarz inequa

Yeh page "toolbox unpack karo" wala page hai. parent note symbols fast fire karta hai: $a$ , $∥ a ∥$ , $\sum$ , $cos θ$ , $t^{2}$ , discriminant. Yahan hum har cheez se pehle milte hain use use karne se, use ek picture se jodte hain, aur kehte hain kyun topic us ke bina nahi chal sakta.

0. Vector kya hota hai? (bilkul pehla object)

Sab kuch yahan se start hota hai, toh hum sirf kagaz ke ek dot ke saath shuru karte hain.

Picture. Apni pencil origin (corner point $(0, 0)$ ) pe rakho. Kisi jagah tak ek arrow draw karo. Woh arrow hi vector hai. Numbers $(3, 4)$ ka matlab hai "3 steps right jao, phir 4 steps upar jao" — arrow ka tip wahan land karta hai.

Figure — Dot product — formula, cosine formula, Cauchy-Schwarz inequality proof

Topic ko yeh kyun chahiye. Dot product ek aisi machine hai jiske inputs do vectors hote hain. Agar tumhare paas arrow-with-tip-coordinates ka crisp mental image nahi hai, toh baad ke har symbol vacuum mein float karte rahenge.

1. Components aur subscript notation $a_{1}, a_{2}, \dots, a_{n}$

Notation padhna.

Neeche wala chota number — $a_{i}$ mein subscript $i$ — sirf ek label hai, ek address. $a_{1}$ pehla component hai, $a_{2}$ doosra. Yeh multiplication nahi hai aur power bhi nahi hai.
Letter $n$ ka matlab hai "jitne bhi components hain." Flat drawing (2D) mein $n = 2$ ; space (3D) mein $n = 3$ ; abstractly $n$ kuch bhi ho sakta hai.

Picture. $a = (3, 4)$ ke liye: $a_{1} = 3$ horizontal reach hai, $a_{2} = 4$ vertical reach hai. Figure s01 dekho — do dashed legs exactly $a_{1}$ aur $a_{2}$ hain.

Topic ko yeh kyun chahiye. Algebraic dot product $a_{1} b_{1} + a_{2} b_{2} + \dots$ hai — tum literally ise bina components ko unke address se naam diye likh hi nahi sakte.

2. Symbol $R^{n}$ — jahaan arrows rehte hain

Picture. $R^{2}$ flat page hai (2 numbers → kagaz pe ek point). $R^{3}$ woh room hai jisme tum baithe ho (3 numbers). Bade $n$ ke liye $R^{n}$ draw nahi kar sakte, lekin algebra bilkul identically kaam karta hai — yahi ise naam dene ka point hai.

Topic ko yeh kyun chahiye. Parent Cauchy–Schwarz bina koi picture draw kiye prove karta hai, precisely isliye taaki woh $R^{n}$ mein survive kare jahan drawing impossible hai. " $R^{n}$ = number-lists ka space" jaanna hi proof ko honest rehne deta hai.

3. Vectors add aur subtract karna (component-wise)

Kisi bhi $\sum$ ya dot se pehle, humein jaanna hai ki do arrows ko combine kaise karte hain. Yeh us waqt use hota hai jab parent $a - b$ aur $a - t b$ likhta hai.

Picture. $a + b$ ek "tip-to-tail" walk hai: $b$ ko $a$ ki tip se start karo; origin se final tip tak ka combined arrow sum hai. Difference $a - b$ woh arrow hai jo $b$ ki tip se $a$ ki tip ki taraf point karta hai (yeh us triangle ki teesri side hai jo woh banate hain).

Topic ko yeh kyun chahiye. Cosine formula us triangle pe derive hoti hai jiske sides $a$ , $b$ , aur $a - b$ hain; Cauchy–Schwarz proof $a - t b$ study karta hai. Dono sirf component-wise subtraction hain — agar arrows subtract nahi kar sakte, toh koi bhi derivation samajh nahi aayegi.

Recall Do vectors subtract karo

$(3, 4) - (1, 2)$ compute karo. ::: $(2, 2)$

4. Summation sign $i = 1 \sum n$

Parent mein yeh sabse scary-looking symbol hai, aur iska matlab kuch bilkul simple hai.

Isse slowly padhna.

Bada Greek $\sum$ (capital sigma, "Sum" ke liye ek "S") = "add up."
Neeche $i = 1$ = counter 1 se start karo.
Upar $n$ = counter $n$ reach karne par ruko.

Toh $\sum_{i = 1}^{n} a_{i} b_{i} = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} .$ Yeh sirf shorthand hai ek lambi "plus" chain ke liye — kuch aur nahi.

Topic ko yeh kyun chahiye. $a_{1} b_{1} + a_{2} b_{2} + \dots + a_{100} b_{100}$ likhna exhausting hai. $\sum$ poore pattern ko ek compact symbol mein pack karta hai. Dot product ki definition ek single $\sum$ hai.

Recall Yeh sum khud unpack karo

$i = 1 \sum 3 a_{i} b_{i}$ poora likh ke dikhao. ::: $a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$

5. Dot operator " $\cdot$ " — machine khud

Hum iska baar baar reference kar rahe hain, toh symbol ko pin down karte hain pehle, kaafi use karne se.

Symbol padhna. Wahi chota dot do plain numbers ke beech (jaise $3 \cdot 4$ ) ordinary multiplication mean karta hai. Do bold vectors ke beech iska matlab "upar wala sum" hai. Same dot, lekin dono taraf ki cheez ka type batata hai ki woh kaunsa kaam kar raha hai.

Picture. Component $a_{1}$ ko $b_{1}$ ke paas slide karo aur multiply karo; har axis ke liye aisa hi karo; un saare products ko ek pile mein daalo aur add karo. Ek vector ek haath mein, ek doosre mein, ek number bahar.

Topic ko yeh kyun chahiye. Yahi toh poora subject hai. Ab tak ke har symbol (components, $\sum$ ) isliye exist karte the taaki yeh ek line likhi ja sake. Jab Section 6 $a \cdot a$ likhega, iska matlab sirf " $a$ ko is machine ke dono slots mein daalo" hai.

Recall Machine ek baar chalao

$(1, 2, 2) \cdot (2, 0, - 1)$ compute karo. ::: $1 \cdot 2 + 2 \cdot 0 + 2 \cdot (- 1) = 0$

6. Length / magnitude $∥ a ∥$ aur Pythagorean picture

Picture — yeh Pythagoras hai. $a = (3, 4)$ ke liye, arrow ek right triangle ka hypotenuse hai jiske legs components $3$ aur $4$ hain. Toh $∥ a ∥ = 3^{2} + 4^{2} = 9 + 16 = 25 = 5.$ Generally $∥ a ∥ = a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2}$ .

Square root kyun use karein, aur yeh tool kyun? Humein ek distance chahiye, aur Pythagorean theorem woh ek rule hai jo "sideways reach + upward reach" ko "straight-line reach" mein convert karta hai. Square root squares ko undo karta hai taaki hum wapas ordinary length units mein land karein.

Dot product se bridge. Section 5 ki dot machine ke dono slots mein $a$ daalo. Uski definition se, $a \cdot a = \sum_{i = 1}^{n} a_{i} a_{i} = a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2}$ — jo exactly wahi hai jo upar square root ke neeche baitha hai. Isliye $∥ a ∥ = a_{1}^{2} + \dots + a_{n}^{2} = a \cdot a .$ Length aaine mein dekh raha dot product hai: yeh root ke neeche $\sum_{i} a_{i} a_{i}$ sum se zyada kuch nahi. Yeh link baad mein har jagah use hoti hai, toh isse lock in karo. Vector projection dekho jahan length + direction alag kiye jaate hain.

7. Angle $θ$ aur cosine $cos θ$

Picture. $θ$ angle do arrows ke beech draw karo. Jaise ek arrow ko lined-up se perpendicular ki taraf swing karo, $cos θ$ smoothly $1$ se $0$ tak slide karta hai; aur opposite tak swing karo, $- 1$ tak slide karta hai.

Yeh tool kyun, koi aur kyun nahi? Humein ek single number chahiye jo aligned hone par bada ho, perpendicular hone par zero ho, opposed hone par negative ho — aur yeh exactly cosine ka behavior hai. Koi aur elementary function is "sameness dial" ko itne cleanly match nahi karta. Isliye geometric formula $a \cdot b = ∥ a ∥∥ b ∥ cos θ$ mein $cos$ use hota hai, $sin$ nahi. (Sibling Cross product mein $sin$ use hota hai precisely isliye kyunki woh perpendicular part measure karta hai instead.)

Woh key fact jo hum reuse karenge. Kyunki $cos θ$ kabhi $[- 1, 1]$ nahi chod sakta, humeshaa $∣ cos θ ∣ \leq 1$ hota hai. Woh inequality Cauchy–Schwarz ka geometric heart hai. Law of cosines dekho — woh exactly woh tool hai jo parent cosine formula derive karne ke liye use karta hai.

Recall Dial check karo

$cos θ$ kya hoga jab do vectors perpendicular hain? ::: $0$

8. Scalar $λ$ , "scalar multiple", aur parallel hona

Picture. $2 b$ , $b$ se do guna lamba hai, same direction mein. $- \frac{1}{2} b$ aadha lamba hai, opposite direction mein point karta hai.

Topic ko yeh kyun chahiye. Cauchy–Schwarz ka equality case (" $∣ a \cdot b ∣ = ∥ a ∥∥ b ∥$ exactly jab $a, b$ parallel hain") scalar multiples ke baare mein ek statement hai. Aur poora discriminant proof vector $a - t b$ study karta hai ( $b$ ko $t$ se scalar-multiply karo, phir Section 3 ki tarah component-wise subtract karo) jaise $t$ vary karta hai.

9. Quadratic $A t^{2} + B t + C$ aur uska discriminant

Cauchy–Schwarz proof ek vector problem ko school-level parabola mein convert kar deta hai. Tumhe woh parabola chahiye.

Picture. Ek upward U ke liye jo kabhi zero se neeche nahi dip karta, graph at most axis ko sirf touch kar sakta hai — toh iska at most ek root hai — toh $B^{2} - 4 A C \leq 0$ . Woh single inequality algebraic Cauchy–Schwarz proof ka poora engine hai.

Yeh tool kyun? Proof $f (t) = ∥ a - t b ∥^{2}$ build karta hai, jo ek squared length hai isliye woh kabhi negative nahi ho sakta. "Upward parabola jo kabhi negative nahi" ⟹ discriminant $\leq 0$ , aur us inequality ko rearrange karna hi Cauchy–Schwarz hai. Discriminant bridge hai "geometry $\geq 0$ kehti hai" se "algebra ek inequality spit out karta hai" tak.

ek arrow jisme length aur direction ho, numbers ki list jaise

(3, 4)

ke roop mein likha jaata hai

The subscript in

a_{i}

means

ek address/label (

i

-th component), NOT a power

R^{n}

n

real numbers ki saari lists ka space (saare

n

-component arrows)

To subtract vectors you

matching components subtract karo:

(a_{1} - b_{1}, \dots, a_{n} - b_{n})

The sign

i = 1 \sum n

tells you to

expression add karo jaise counter

i

1, 2, \dots, n

run karta hai

The dot operator

a \cdot b

means

matching components multiply karo aur add karo:

\sum_{i} a_{i} b_{i}

(ek single number)

∥ a ∥

means

arrow ki length,

a_{1}^{2} + \dots + a_{n}^{2}

ke barabar (Pythagoras)

The length in dot-product form is

∥ a ∥ = a \cdot a

kyunki

a \cdot a = \sum_{i} a_{i}^{2}

cos θ

ranges over

- 1

+ 1

tak; aligned hone par

+ 1

, perpendicular hone par

0

, opposite hone par

- 1

The angle

θ

is undefined when

koi ek vector zero vector

0

ho (koi direction nahi)

Why cosine (not sine) appears in the dot product

cosine "sameness dial" hai: aligned hone par bada, perpendicular hone par zero

A scalar multiple

λ b

arrow ko rotate kiye bina stretch ya flip karta hai; same-direction line

Two vectors are parallel when

a = t b

kisi scalar

t

ke liye

The discriminant of

A t^{2} + B t + C

B^{2} - 4 A C

; yeh

\leq 0

hota hai jab ek upward parabola kabhi zero se neeche nahi dip karta

∣ x ∣

∥ a ∥

single bars = number ki absolute value; double bars = vector ki length

a ⊥ b

means

a

b

ke perpendicular hai (angle

9 0^{\circ}

, dot product

0

)

Foundations — Dot product — formula, cosine formula, Cauchy-Schwarz inequality proof

0. Vector kya hota hai? (bilkul pehla object)

1. Components aur subscript notation $a_{1}, a_{2}, \dots, a_{n}$

2. Symbol $R^{n}$ — jahaan arrows rehte hain

3. Vectors add aur subtract karna (component-wise)

4. Summation sign $i = 1 \sum n$

5. Dot operator " $\cdot$ " — machine khud

6. Length / magnitude $∥ a ∥$ aur Pythagorean picture

7. Angle $θ$ aur cosine $cos θ$

8. Scalar $λ$ , "scalar multiple", aur parallel hona

9. Quadratic $A t^{2} + B t + C$ aur uska discriminant

10. Perpendicular symbol $⊥$ aur doosre shorthand

Foundations topic ko kaise feed karti hain

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