Projection of vectors

4.5.4 · Maths › Linear Algebra (Full)

WHAT we are computing

HOW to derive the formula (from scratch)

Hum chahte hain ki $∣ a ∣ cos θ$ ko coordinates ke terms mein nikaalein, bina protractor ke.

Step 1 — Dot product definition se shuru karo. $a \cdot b = ∣ a ∣ ∣ b ∣ cos θ$ Yeh step kyun? Dot product ek aisa tool hai jisme already secretly $cos θ$ chhupa hua hai, isliye yeh humein angle eliminate karne deta hai.

Step 2 — Shadow length $∣ a ∣ cos θ$ solve karo. Dono sides ko $∣ b ∣$ se divide karo: $∣ a ∣ cos θ = \frac{a \cdot b}{∣ b ∣}$ Yeh step kyun? Left side exactly scalar projection hai. Humne use isolate kar liya.

Step 3 — Length ko ek vector banao. Length ko unit direction $\hat{b} = b /∣ b ∣$ se multiply karo: $proj_{b} a = (\frac{a \cdot b}{∣ b ∣}) \frac{b}{∣ b ∣} = \frac{a \cdot b}{∣ b ∣ ^{2}} b = \frac{a \cdot b}{b \cdot b} b$ Yeh step kyun? Vector = (length) × (direction). Hum signed length ke saath direction $\hat{b}$ attach kar rahe hain.

Step 4 — Bacha hua (orthogonal) part. $a$ ka jo bhi hissa $b$ ke along nahi hai woh uske perpendicular hoga: $a_{⊥} = a - proj_{b} a$ Check karo: $a_{⊥} \cdot b = a \cdot b - \frac{a \cdot b}{b \cdot b} (b \cdot b) = 0$ . ✓ Toh $a = proj_{b} a + a_{⊥}$ , $a$ ko parallel + perpendicular pieces mein split karta hai.

WHY the sign matters

$cos θ$ negative hota hai jab $θ > 9 0^{\circ}$ . Tab scalar projection negative hoti hai, aur vector projection $b$ ke opposite direction mein point karta hai. Yeh correct hai: shadow "peeche" wali side par padti hai.

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bachche ko samjhao

Ek jhuki hui laathi par seedha torch shine karo. Zameen par jo flat shadow banta hai woh batata hai "laathi us zameen ki direction mein kitni dur tak jaati hai." Us shadow ki length hi projection hai. Agar laathi doosri taraf jhuke, toh uski shadow peeche jaati hai — wohi minus sign hai. Length nikaalane ke liye hum angles measure karne ki jagah ek jaldi multiply-and-add trick (dot product) use karte hain.

Active recall

Scalar projection of

a

onto

b

formula?

\frac{a \cdot b}{∣ b ∣} = a \cdot \hat{b}

Vector projection of

a

onto

b

formula?

\frac{a \cdot b}{b \cdot b} b

Why does the vector form divide by

b \cdot b

, not

∣ b ∣

∣ b ∣

length set karta hai, ek aur

b

ko unit direction banata hai

\Rightarrow ∣ b ∣^{2}

What does a negative scalar projection mean?

Angle obtuse hai; shadow

b

ke opposite point karta hai.

How do you get the part of

a

perpendicular to

b

a_{⊥} = a - proj_{b} a

proj_{b} a = proj_{a} b

Nahi — projection symmetric nahi hai; direction alag hoti hai.

Projection of

(3, 4)

onto the

x

-axis

(1, 0)

(3, 0)

, scalar

3

Which trig function appears in scalar projection and why?

cos θ

— yeh

b

ke along adjacent component deta hai.

Connections

Dot product — projection ke peeche ka engine.
Unit vectors — $\hat{b}$ direction provide karta hai.
Orthogonal decomposition — $a$ ko parallel + perpendicular mein split karna.
Gram-Schmidt process — orthonormal bases banane ke liye repeated projection.
Least squares regression — data ko column space par project karna.
Work done by a force — physics application: $W = F \cdot d$ force ka displacement ke along projection use karta hai.