Foundations — Projection of vectors

4.5.4 · D1 · Maths › Linear Algebra (Full) › Projection of vectors

Is page par assume kiya gaya hai ki tumne kuch bhi nahi dekha. Hum har ek symbol ko naam denge jo parent Projection of vectors note use karta hai, uske peeche ka picture draw karenge, aur batayenge ki ye topic uske bina kyun exist nahi kar sakta. Upar se neeche padho; har block agla block earn karta hai.

1 — Vector kya hota hai? Symbol $\vec{a}$

Ek plain number jaise $5$ ki sirf size hoti hai. Ek arrow ki size hoti hai aur ye bhi ki wo kis taraf point karta hai. Wahi "kis taraf" projection ke exist hone ki wajah hai — hum directions compare kar rahe hain.

Picture mein burnt-orange arrow origin (corner point $O$) se shuru hota hai aur apni tip par khatam hota hai. Arrow ki length batati hai ki wo kitna bada hai; arrow ka tilt uski direction hai.

2 — Coordinates: $\vec{a}=(3,4)$ likhna

• $\vec{a}=(3,4)$ ka matlab hai: $O$ se, 3 right, phir 4 upar jaao. Us landing spot ka arrow hi $\vec{a}$ hai.
• Ek three-dimensional vector sirf "depth" ke liye ek teesra number add karta hai: $\vec{a}=(2,1,2)$ (parent ke Example 4 mein use hota hai).

3 — Vector ki length: symbol $|\vec{a}|$

Iska formula kahan se aata hai? Figure s02 dobara dekho: components $a_x$ aur $a_y$ ek right triangle ki do choti sides banate hain, aur arrow $\vec{a}$ slanted long side (hypotenuse) hai. Pythagoras — "do legs squared, add karo, phir square root lo" — length deta hai:

$|\vec{a}|=\sqrt{a_x^2+a_y^2}$

Topic ko ye kyun chahiye. Scalar projection $|\vec{a}|\cos \theta$ literally $|\vec{a}|$ se shuru hota hai — "shadow kitni door tak jaata hai" iske baare mein baat karna impossible hai bina length ke notion ke. Aur formula mein $|\vec{b}|$ se division hi rokta hai ki answer sirf isliye bade na ho kyunki tumne $\vec{b}$ lamba draw kiya.

4 — Do arrows ke beech ka angle: $\theta$

Figure teen cases dikhata hai jinke liye tumhe taiyaar rehna chahiye:

• Acute ($\theta <90^{\circ }$): arrows roughly ek direction mein agree karte hain. Shadow aage girti hai.
• Right angle ($\theta =90^{\circ }$): arrows perpendicular hain. Shadow ki length zero hoti hai — $\vec{a}$ ka koi bhi hissa $\vec{b}$ ki taraf point nahi karta.
• Obtuse ($\theta >90^{\circ }$): arrows disagree karte hain, alag-alag direction mein point karte hain. Shadow peeche ki taraf girti hai — yahi wajah hai parent ka minus sign.

Teeno ko yaad rakho: parent ka "sign matters" section kuch nahi hai bas in teen pictures ke siva.

5 — Cosine: $\cos \theta$ kyun, koi aur function kyun nahi

Cosine ka sign — har case cover karte hue:

• $\cos \theta >0$ for $\theta <90^{\circ }$ → shadow aage (positive).
• $\cos \theta =0$ at $\theta =90^{\circ }$ → shadow zero.
• $\cos \theta <0$ for $\theta >90^{\circ }$ → shadow peeche (negative).

Toh ek single number $\cos \theta$ dono kitna aur kis taraf carry karta hai — isliye parent insist karta hai ki tum kabhi iska absolute value mat lo.

6 — Dot product: symbol $\vec{a}\cdot \vec{b}$

7 — Unit vectors: symbol $\hat{b}$

Isliye bhi vector projection $|\vec{b}|$ se do baar divide karta hai: ek baar $\hat{b}$ ke andar direction normalise karne ke liye, ek baar length lene ke liye. Wo doubled length $\vec{b}\cdot \vec{b}=|\vec{b}{|}^2$ likha jaata hai.

8 — Perpendicular / orthogonal: symbol ${\vec{a}}_{\perp }$ aur "$\cdot =0$"

Zero ka matlab perpendicular kyun hai? Geometry form $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos 90^{\circ }$ se, aur $\cos 90^{\circ }=0$. Toh dot product exactly tab vanish hota hai jab wo right angle par milte hain. Symbol ${\vec{a}}_{\perp }$ (padho "a-perp") $\vec{a}$ ka leftover hissa hai jab tum uski shadow remove kar dete ho — ye $\vec{b}$ ke saath clean right angle par point karta hai. Ye Orthogonal decomposition ka seed hai.

Prerequisite map

Equipment checklist

Khud test karo — har line prompt ::: answer hai.

Connections

• Dot product — wo number-machine jo $\cos \theta$ chhupata hai; projection ko protractor ki zaroorat na hone ki wajah.
• Unit vectors — pure direction $\hat{b}$ provide karta hai jo shadow carry karta hai.
• Orthogonal decomposition — jahan ${\vec{a}}_{\perp }$ aage jaata hai.
• Projection of vectors — parent topic jisko ye foundations feed karte hain.