Visual walkthrough — Projection of vectors

4.5.4 · D2 · Maths › Linear Algebra (Full) › Projection of vectors

Step 0 — Do arrows aur ek floor

Kisi bhi formula se pehle, humein yeh agree karna hoga ki hum dekh kya rahe hain.

Ek arrow (ek vector) bas ek length hoti hai jo kisi direction mein point karti hai. Hum donon ko same point $O$ se draw karte hain:

• $\vec{a}$ — woh arrow jiska shadow lena hai.
• $\vec{b}$ — woh arrow jis par shadow daalna hai. $\vec{b}$ ko ek floor-direction samjho, ek aisi line jis par shadows daale jaate hain.

Kisi letter ke upar chhota sa $\stackrel{⃗}{}$ symbol ka matlab bas yeh hai ki "yeh ek arrow hai, koi plain number nahi".

Step 1 — Shadow daalo: ek right triangle saamne aata hai

KYA. $\vec{b}$ ki line par seedha neeche light shine karo. $\vec{a}$ ki tip us line par kahin land karti hai. $O$ se us landing point tak ka segment hi shadow hai. Iske signed length ko $s$ bol lo.

KYUN. "Seedha neeche" ka matlab hai ki light rays floor par $90^{\circ }$ par girti hain. Woh perpendicular drop hi ek aisi cheez hai jo ek clean triangle banata hai — aur right triangle ek aisi shape hai jiske sides ko hum relate karna jaante hain.

PICTURE. Red dashed line dekho jo $\vec{a}$ ki tip se neeche gir rahi hai: woh $\vec{b}$ ki line ko ek perfect right angle par milti hai (chhota sa square). Isse hume ek right triangle milta hai:

• slanted side (hypotenuse) khud $\vec{a}$ hai, jiska length $|\vec{a}|$ hai;
• base jo $\vec{b}$ ke along hai, woh shadow hai jiska length $s$ hai;
• $O$ par angle jo $\vec{a}$ aur $\vec{b}$ ke beech hai, use hum $\theta$ (Greek letter "theta", us opening angle ka bas ek naam) bolte hain.

Vertical bars $|\vec{a}|$ ka matlab hai $\vec{a}$ ki length — ek plain positive number, jaise arrow ko ruler se measure karna.

Step 2 — $\cos \theta$ kyun, aur kuch nahi, shadow deta hai

KYA. Hamare right triangle mein, shadow $s$ woh side hai jo angle $\theta$ ke saath (adjacent side) hai, aur $|\vec{a}|$ hypotenuse hai. Hum claim karte hain ki $s=|\vec{a}|\cos \theta .$

Yeh tool KYUN? Hume ek aisi machine chahiye jo angle $\theta$ aur hypotenuse $|\vec{a}|$ leke adjacent side nikale. Woh machine cosine hai. Right triangle par iske definition se hi: $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{s}{|\vec{a}|}.$ Humne sine nahi chuna (woh opposite side deta hai, shadow ki height floor se upar — jise hum throw away kar rahe hain) aur tangent nahi chuna (woh donon sides ka ek aisa ratio hai jo hypotenuse ki actual length ignore karta hai). Cosine perfect fit hai: yeh measure karta hai hypotenuse ka kitna hissa floor par lean hone ke baad bachta hai.

PICTURE. Jaise $\theta$ $0$ se badhta hai, arrow floor se zyada door lean karta hai aur green shadow chhoti hoti jaati hai — exactly wahi jo $\cos \theta$ karta hai (woh $1$ se $0$ ki taraf girta hai). $\theta =90^{\circ }$ par arrow seedha upar point karta hai, shadow gaayab ho jaata hai, aur $\cos 90^{\circ }=0$. Perfect match.

Multiply karke: $s=|\vec{a}|\cos \theta$. Yahan $|\vec{a}|$ arrow ki poori length hai aur $\cos \theta$ us length ka woh fraction hai jo $\vec{b}$ ke along padi hai.

Step 3 — Shadow ka sign: back-side case cover karo

KYA. Kya ho agar $\theta$ $90^{\circ }$ se bada ho — $\vec{a}$ $\vec{b}$ se door lean kare? Tab landing point $O$ ki opposite side par hogi, aur hum kehte hain shadow length $s$ negative hai.

KYUN. $\cos \theta$ $\theta <90^{\circ }$ ke liye positive hota hai, exactly $90^{\circ }$ par zero, aur $\theta >90^{\circ }$ ke liye negative. Toh $s=|\vec{a}|\cos \theta$ automatically negative ho jaata hai. Yeh bug nahi, feature hai: minus sign record karta hai ki shadow $O$ ki kis side par gira.

PICTURE. Teen cases side by side:

• $\theta <90^{\circ }$ (blue): shadow $\vec{b}$ ke aage point karta hai, $s>0$.
• $\theta =90^{\circ }$: $\vec{a}$ perpendicular hai, shadow ki length zero hai, $s=0$.
• $\theta >90^{\circ }$ (red): shadow peeche point karta hai, $s<0$.

Step 4 — Protractor se bachne ke liye dot product use karo

KYA. Formula $|\vec{a}|\cos \theta$ honest hai lekin sirf coordinates ke saath useless hai: koi $\theta$ nahi deta. Hum ise dot product use karke swap karte hain.

Yeh tool KYUN? Dot product do facts se define hota hai jo secretly ek hi hain: $\vec{a}\cdot \vec{b}=a_x{b}_x+a_y{b}_y(\text{matching parts multiply karo, add karo})=|\vec{a}||\vec{b}|\cos \theta .$ Right-hand side mein pehle se hi woh $\cos \theta$ hai jo chahiye, donon lengths ke saath glued. Toh dot product woh ek tool hai jo ek angle ko pure arithmetic ke andar chhupa leta hai — yeh humein $\cos \theta$ ka effect bina angle measure kiye compute karne deta hai.

PICTURE. Dot product bar: coordinates left slot mein jaate hain, geometric meaning right slot se nikalti hai — do alag routes se same number.

Ab shadow isolate karo. Hume akela $|\vec{a}|\cos \theta$ chahiye, toh $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$ ke donon sides ko $|\vec{b}|$ se divide karo: $\underset{\text{pure arithmetic}}{\underbrace{\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}}}=\underset{\text{shadow }s}{\underbrace{|\vec{a}|\cos \theta }}.$ Har symbol: $\vec{a}\cdot \vec{b}$ multiply-and-add number hai, $|\vec{b}|$ $\vec{b}$ ki length hai jise hum divide karte hain kyunki dot product $|\vec{b}|$ ka ek extra factor le aaya tha jo humne maanga nahi tha.

Step 5 — Length se arrow tak

KYA. Ab tak $s=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$ bas ek number hai. Vector projection ek actual arrow hai: uski woh length hai aur woh $\vec{b}$ ke along point karta hai.

KYUN. Koi bhi vector (kitna lamba) × (kis direction mein) hota hai. Hamare paas length $s$ hai. "Kis direction mein" unit vector $\hat{b}$ hai — $\vec{b}$ ki direction jis par se uski length hata di gayi hai. Inhein attach karte hain: ${\text{proj}}_{\vec{b}}\vec{a}=s\hat{b}=\left(\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}\right)\frac{\vec{b}}{|\vec{b}|}=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}{|}^2}\vec{b}=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}\vec{b}.$

Denominator dekho: ek $|\vec{b}|$ Step 4 mein length nikaalte waqt aaya, doosra $|\vec{b}|$ $\vec{b}$ ko unit direction mein convert karne se aaya. Milke $|\vec{b}|\cdot |\vec{b}|=|\vec{b}{|}^2=\vec{b}\cdot \vec{b}$ (kyunki $\vec{b}\cdot \vec{b}=|\vec{b}{|}^2\cos 0=|\vec{b}{|}^2$).

PICTURE. Green arrow finished vector projection hai, $\vec{b}$ ki line par length $s$ ke saath baitha hai, aur $\vec{b}$ khud neeche unit length tak scaled dikhaya gaya hai.

Step 6 — Bacha hua piece zaroor perpendicular hoga

KYA. $\vec{a}$ mein se shadow-arrow minus karo. Jo bachta hai woh $\vec{a}$ ka woh part hai jiska $\vec{b}$ ke along koi direction nahi tha: ${\vec{a}}_{\perp }=\vec{a}-{\text{proj}}_{\vec{b}}\vec{a}.$

KYUN. Construction se hi projection ne $\vec{a}$ ka poora along-$\vec{b}$ content absorb kar liya. Jo bacha woh sirf $\vec{b}$ se right angles par hi point kar sakta hai. Ise prove karte hain — bacha hua piece $\vec{b}$ ke saath dot product lete hain: ${\vec{a}}_{\perp }\cdot \vec{b}=\vec{a}\cdot \vec{b}-\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}(\vec{b}\cdot \vec{b})=\vec{a}\cdot \vec{b}-\vec{a}\cdot \vec{b}=0.$ Zero dot product ka matlab hai donon arrows $90^{\circ }$ par milte hain. Toh ${\vec{a}}_{\perp }\perp \vec{b}$. ✓

PICTURE. $\vec{a}$ (orange) ab do perpendicular arrows ka exact sum hai: $\vec{b}$ ke along green shadow plus red perpendicular remainder. Yeh ek right triangle banate hain jiska hypotenuse $\vec{a}$ hai — wahi triangle Step 1 se, ab vectors se label kiya gaya.

Yeh split $\vec{a}={\text{proj}}_{\vec{b}}\vec{a}+{\vec{a}}_{\perp }$ Orthogonal decomposition hai, jo Gram-Schmidt process aur Least squares regression ka engine hai.

Step 7 — Degenerate case: kya ho agar $\vec{b}=\vec{0}$?

KYA. Agar $\vec{b}$ zero vector hai (ek point, koi direction nahi), tab $\vec{b}\cdot \vec{b}=0$ aur formula humse zero se divide karne ko kehta hai.

KYUN yeh break hona chahiye. Shadow ko giraane ke liye ek direction chahiye. Zero vector kahin nahi point karta, toh "$\vec{a}$ ka nothing par shadow" ka koi matlab nahi — maths sahi tarah se refuse karta hai (division by zero) rather than ek fake answer invent kare.

PICTURE. $\vec{b}$ origin par collapse ho gaya: shadow giraane ke liye koi line nahi hai, aur projection undefined hai.

Ek picture ka summary

Ek hi canvas par sab kuch: right triangle (Step 1–2), dot product jo angle replace karta hai (Step 4), green shadow-arrow (Step 5), aur red perpendicular leftover (Step 6).

Active recall

Connections

• Projection of vectors — parent topic jiska yeh walkthrough derive karta hai.
• Dot product — multiply-and-add engine jo $\theta$ eliminate karta hai.
• Unit vectors — $\hat{b}$ Step 5 mein direction supply karta hai.
• Orthogonal decomposition — Step 6 ka parallel + perpendicular split.
• Gram-Schmidt process — orthonormalise karne ke liye repeated projection.
• Least squares regression — data ko subspace par project karna.
• Work done by a force — $W=\vec{F}\cdot \vec{d}$ force ka displacement ke along projection use karta hai.