Dot product — formula, geometric meaning, work calculation

1.1.11 · Physics › Measurement, Vectors & Kinematics

1. Yeh hai kya? (Definition)

HAR piece ka matlab:

$∣ A ∣, ∣ B ∣$ = lengths (magnitudes) hain.
$cos θ$ = "alignment factor" hai: parallel hone par $1$ , perpendicular hone par $0$ , opposite hone par $- 1$ .

2. $cos θ$ kyun aata hai? (Scratch se Derivation)

Toh dot product ko define karte hain (A ka B par projection) × (B ki length) ke roop mein: $A \cdot B = (∣ A ∣ cos θ) ∣ B ∣ = ∣ A ∣∣ B ∣ cos θ .$

Ab component form derive karo — yeh prove karne ke liye ki dono definitions agree karti hain.

Basis unit vectors $\hat{i}, \hat{j}, \hat{k}$ use karo. Yeh mutually perpendicular aur unit length hain: $\hat{i} \cdot \hat{i} = ∣ \hat{i} ∣∣ \hat{i} ∣ cos 0 = 1, \hat{i} \cdot \hat{j} = (1) (1) cos 9 0^{\circ} = 0.$ Yeh step kyun? Yeh seedha geometric definition se aata hai jab $θ = 0$ ya $9 0^{\circ}$ hota hai.

$A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$ likho aur similarly $B$ ke liye. Distributivity use karke expand karo (jo dot product follow karta hai — properties dekho): $A \cdot B = A_{x} B_{x} (\hat{i} \cdot \hat{i}) + A_{x} B_{y} (\hat{i} \cdot \hat{j}) + \dots$ Yeh step kyun? A ka har term B ke har term se multiply karo, jaise brackets expand karte hain.

Saare "mixed" terms ( $\hat{i} \cdot \hat{j}$ etc.) $0$ hain; "matched" terms ( $\hat{i} \cdot \hat{i}$ ) $1$ hain: $A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$ Yeh step kyun? Sirf like-with-like products survive karte hain — yahi poora trick hai.

3. Geometric meaning — sign padho!

Figure — Dot product — formula, geometric meaning, work calculation

Intuition Sign tumhe relationship batata hai

Angle $θ$	$cos θ$	$A \cdot B$	Matlab
$0^{\circ}$	$+ 1$	$+	\vec A
$0^{\circ} < θ < 9 0^{\circ}$	$+$	positive	partly aligned
$9 0^{\circ}$	$0$	$0$	perpendicular
$9 0^{\circ} < θ < 18 0^{\circ}$	$-$	negative	opposing
$18 0^{\circ}$	$- 1$	$-	\vec A

Key fact: $A \cdot B = 0$ (non-zero vectors ke saath) $⟺$ woh perpendicular hain.

4. Properties (har ek justified)

5. Work dot product kyun use karta hai

$θ = 0$ : force motion ke along → $W = F d$ (maximum, jaise aage kheenchna).
$θ = 9 0^{\circ}$ : force ⟂ motion → $W = 0$ (jaise horizontally move karte puck par gravity; circular motion mein centripetal force).
$θ = 18 0^{\circ}$ : force motion ko oppose karta hai → $W < 0$ (jaise friction).

6. Worked examples

7. Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum ek toy car push kar rahe ho. Agar tum seedha aage push karo, tumhari saari push help karti hai use jaane mein. Agar tum thoda sideways push karo, sirf teri push ka ek hissa help karta hai — sideways wala hissa waste ho jaata hai. Dot product ek machine hai: tum do arrows daalo aur woh batata hai "pehla arrow doosre ke same direction mein kitna jaata hai." Jab woh 'help karne wali' amount ko car ke kitni door move hui se multiply karte hain, tumhe work milta hai — woh asli effort jo car ko move kiya. Agar tum seedha neeche push karo jabki car aage roll kar rahi ho, tum isse zero help karte ho — aur dot product exactly $0$ deta hai. Smart machine!

Connections

Vectors — components and unit vectors
Cross product — area and torque (woh "perpendicular" partner)
Work-Energy Theorem (work kinetic energy change karta hai)
Projection of a vector
Circular motion (centripetal force ⟂ velocity ⇒ koi work nahi karta)
Trigonometry — cosine and components

Flashcards

Dot product kya return karta hai — scalar ya vector?

Ek scalar (single number); isliye isse scalar product kehte hain.

Dot product ka geometric formula?

A \cdot B = ∣ A ∣∣ B ∣ cos θ

Dot product ka component formula?

A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}

A \cdot B = 0

ka kya matlab hai (non-zero vectors)?

Woh perpendicular hain (

θ = 9 0^{\circ}

Dot product mein

cos θ

kyun aata hai?

Yeh ek vector ka doosre par projection (shadow) pick out karta hai.

Do vectors ke beech angle ka formula?

cos θ = \frac{A \cdot B}{∣ A ∣∣ B ∣}

A \cdot A

kya hai?

∣ A ∣^{2}

(kyunki

θ = 0

hai).

Constant force se kiya gaya work?

W = F \cdot d = F d cos θ

, unit joule.

Force ka work zero kab hota hai?

Jab force displacement ke perpendicular ho (

θ = 9 0^{\circ}

Centripetal force ka work zero kyun hota hai?

Yeh har instant par velocity/displacement ke perpendicular hota hai.

Negative work ka physically kya matlab hai?

Force object se energy remove karta hai (jaise friction).

(3, 4, 0) \cdot (2, - 1, 2)

compute karo.

6 - 4 + 0 = 2