Vector representation — magnitude, direction, components

1.1.7 · Physics › Measurement, Vectors & Kinematics

1. Ek hi arrow ko describe karne ke do tarike

KYU do forms? Polar form woh hai jaise nature data deti hai ("50 N at 30°"). Component form woh hai jaise hum calculate karte hain (hum components add karte hain, arrows nahi). Hum inke beech lagaataar translate karte rehte hain.

Figure — Vector representation — magnitude, direction, components

2. Derivation: magnitude–direction → components

Arrow $A$ ko origin se draw karo. Iske tip se x-axis par ek perpendicular daalo. Isse ek right triangle banta hai:

hypotenuse = $A$ (poora arrow),
origin par angle = $θ$ ,
x ke along wali side $θ$ ke adjacent hai,
y ke along wali side $θ$ ke opposite hai.

Ek right triangle mein cosine aur sine ki definitions se: $cos θ = \frac{adjacent}{hyp} = \frac{A _{x}}{A}, sin θ = \frac{opposite}{hyp} = \frac{A _{y}}{A} .$

Dono solve karne par:

3. Derivation: components → magnitude & direction

Ab ulta chalte hain. Usi right triangle ki legs $A_{x}, A_{y}$ hain. Pythagoras se (legs² ka sum hypotenuse² ke barabar): $A^{2} = A_{x}^{2} + A_{y}^{2} \Rightarrow A = A_{x}^{2} + A_{y}^{2} .$

Angle ke liye, dono component equations ko divide karo: $\frac{A _{y}}{A _{x}} = \frac{A s i n θ}{A c o s θ} = tan θ \Rightarrow θ = tan^{- 1} (\frac{A _{y}}{A _{x}}) .$

Quadrant check kyun? Calculator par $tan^{- 1}$ sirf $(- 90°, 90°)$ mein angles return karta hai. Lekin 2nd ya 3rd quadrant mein point karne wale arrow ka $A_{y} / A_{x}$ ratio wahi hota hai jo 4th ya 1st mein point karne wale ka. Sahi quadrant jaanne ke liye tumhe $A_{x}, A_{y}$ ke signs dekhne padenge.

$A_{x}$ ka sign	$A_{y}$ ka sign	Quadrant	Calculator angle mein fix
+	+	I	kuch nahi
−	+	II	$180°$ add karo
−	−	III	$180°$ add karo
+	−	IV	$360°$ add karo (ya negative rehne do)

4. Unit vector — size hatake sirf direction

$A$ se divide karne par length 1 ho jaati hai jabki har component usi factor se chhota ho jaata hai, isliye direction change nahi hoti. 2D mein: $\hat{A} = (cos θ) \hat{i} + (sin θ) \hat{j}$ .

5. Worked examples

6. Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum apne dost ko directions de rahe ho: "tezi se chalo, us taraf!" Tumne kitni tezi (size) aur kis taraf (direction) dono bata diya — yahi ek vector arrow hai. Lekin "us taraf diagonally chalo" sunke chalna mushkil hai. Toh tum kehte ho: "3 kadam East chalo, phir 4 kadam North." Yeh do simple steps components hain. Dono descriptions ek hi jagah pahunchti hain. Pythagoras tumhe bataata hai ki seedhi-line distance 5 kadam thi, aur path ka slope direction bataata hai. Components bas kisi bhi tirche arrow ka "pehle East phir North" version hain.

7. Active recall

Ek vector kin do quantities ko carry karta hai?

Magnitude (kitna) aur direction (kis taraf).

A

ka x ke along component

A, θ

(+x se angle) ke terms mein?

A_{x} = A cos θ

Y ke along component?

A_{y} = A sin θ

X-component cosine kyun use karta hai?

Angle x-axis se measure hota hai, toh x-side adjacent hai → cosine.

Components se magnitude?

A = A_{x}^{2} + A_{y}^{2}

(legs par Pythagoras).

Components se direction?

θ = tan^{- 1} (A_{y} / A_{x})

, signs se quadrant correction ke saath.

Calculator ke

tan^{- 1}

mein

180°

kab add karna padta hai?

Jab

A_{x} < 0

ho (vector quadrant II ya III mein point kare).

Unit vector kya hota hai aur kaise milta hai?

Ek length-1 vector same direction mein;

\hat{A} = A /∣ A ∣

Kya magnitude negative ho sakti hai?

Nahi,

A \geq 0

; sign info components/angle mein rehti hai.

A = - 3 \hat{i} + 4 \hat{j}

ke components: magnitude aur sahi angle?

A = 5

θ = 127°

(quadrant II).

Do vectors ki magnitudes directly kyun nahi add kar sakte?

Arrows partially cancel ho sakte hain; pehle components add karo, phir magnitude lo.

Connections

Scalars vs Vectors — kyun ek akela number kaafi nahi hai.
Vector Addition — Triangle & Parallelogram Law — components addition ko trivial bana dete hain.
Resolution of Forces — mechanics mein direct application.
Dot Product and Cross Product — components par built hai.
Projectile Motion — velocity ko $v_{x}, v_{y}$ mein split karna.
Unit Vectors i, j, k — component form ke peeche ka basis.

Vector representation — magnitude, direction, components

1. Ek hi arrow ko describe karne ke do tarike

2. Derivation: magnitude–direction → components

3. Derivation: components → magnitude & direction

4. Unit vector — size hatake sirf direction

5. Worked examples

6. Common mistakes (steel-manned)

7. Active recall

Connections

Concept Map

Deep Dive