DO axes kyun? 2D space mein kisi bhi point ko locate karne ke liye tumhe DO pieces of information chahiye: kitna left/right (x-coordinate) aur kitna up/down (y-coordinate).
Yeh saath mein kaise kaam karte hain? x-axis ko "street number" aur y-axis ko ek building mein "floor number" socho. Ordered pair (x, y) ek complete address hai.
Yeh convention kyun? Mathematicians ne agree kiya ki right = positive aur up = positive, kyunki yeh hmare padhne ke tarike se match karta hai (left se right) aur natural intuition se bhi (upar = zyada).
Axes, plane ko chaar regions mein divide karte hain jise quadrants kehte hain, jo top-right se shuru hokar counterclockwise number hote hain.
Counterclockwise numbering kyun? Yeh mathematics mein positive rotation ki direction follow karta hai (baad mein unit circle ki tarah).
Signs yaad karne ka tarika? Quadrant I se shuru karo (dono positive), phir counterclockwise move karo, signs flip hote hain: (+,+) → (−,+) → (−,−) → (+,−).
Yeh topic tumhare baad mein depth mein aayega, lekin chalte hain scratch se derive karte hain ki coordinates kyun matter karte hain:
Problem: A(x₁, y₁) aur B(x₂, y₂) ke beech distance nikalo.
Step 1: Ek right triangle banao
Point C(x₂, y₁) banane ke liye perpendiculars girao.
WHY? Hum Pythagoras use karna chahte hain, jiske liye right angle chahiye.
Step 2: Horizontal leg ki length nikalo
AC = |x₂ − x₁|
WHY? Horizontal distance sirf x-coordinates ka difference hai.
Step 3: Vertical leg ki length nikalo
CB = |y₂ − y₁|
WHY? Vertical distance, y-coordinates ka difference hai.
Step 4: Pythagoras apply karo
AB2=AC2+CB2AB=(x2−x1)2+(y2−y1)2
Yeh kyun kaam karta hai? Coordinate system geometry ko algebra mein badal deta hai. Cartesian plane ke orthogonal axes Pythagoras ko applicable banate hain.
Socho tum ek treasure hunt video game khel rahe ho. Map ek badi grid hai jisme beech mein do lines cross karti hain — ek left-right jaati hai (jaise kitaab padhna) aur ek upar-neeche (jaise seedhiyan chadhna).
Crossing point ko "home base" ya origin kehte hain. Har treasure ka ek address hai jo do numbers se bana hai: kitne kadam right (ya left agar negative hai) aur kitne kadam upar (ya neeche agar negative hai).
Jab tum likhte ho (3, 5), iska matlab hai "Home se 3 kadam right jao, PHIR 5 kadam upar." Order matter karta hai! Agar tum 5 right aur 3 upar jao, toh alag treasure milega.
Map 4 zones mein divide hai:
Zone 1 (top-right): Dono numbers positive — dhundhna aasaan
Zone 2 (top-left): Pehla number negative (left jao), doosra positive (upar jao)
Zone 3 (bottom-left): Dono negative — dungeon ki gehraai mein
Distance formula — directly Pythagorean theorem se coordinate differences use karke bana hai
Midpoint formula — center dhundhne ke liye coordinates ka average use karta hai
Section formula — kisi bhi ratio division ke liye midpoint ko generalize karta hai
Slope of a line — y ka x ke respect mein rate of change, pure coordinate geometry
Equation of a line — coordinates use karke geometric lines ki algebraic representation
Colinearity — same line par teen points, coordinates se test hota hai
Area of triangle — sirf coordinates se determinants use karke calculate hota hai
Polar coordinates — (x, y) ki jagah distance aur angle use karne wala alternative system
Complex plane — Cartesian plane ko imaginary numbers ko y-axis banake extend karta hai
Vectors — ordered pairs direction aur magnitude represent kar sakte hain
#flashcards/maths
Cartesian plane mein do axes kya hain aur yeh kya represent karte hain? :: x-axis (horizontal) left-right position represent karta hai, aur y-axis (vertical) up-down position. Yeh perpendicular number lines hain jo origin O(0,0) par milti hain.
Ordered pair kya hota hai aur order kyun important hai?
Ek ordered pair (x, y) kisi point ki position specify karta hai jahan x horizontal displacement hai aur y vertical displacement, origin se. Order matter karta hai kyunki (3,5) aur (5,3) alag points hain — pehla coordinate HAMESHA x (horizontal) hota hai.
Chaar quadrants ke liye sign conventions kya hain?
(a, 0) x-axis par hota hai aur (0, b) y-axis par. Axes par ke points kisi quadrant mein nahi aate kyunki ek coordinate zero hota hai.
Point P(−3, 4) kaise plot karte hain?
Origin O(0,0) se shuru karo, 3 units LEFT jao (negative x), phir 4 units UP jao (positive y). Point Quadrant II mein hai.
Points A(x₁, y₁) aur B(x₂, y₂) ke beech distance formula kya hai?
d=(x2−x1)2+(y2−y1)2. Yeh Pythagoras theorem se aata hai jo horizontal aur vertical differences se bane right triangle par apply hota hai.
Axes ko "Cartesian" kyun kehte hain?
René Descartes ke naam par, jinhone coordinate system banaya algebra aur geometry ko connect karne ke liye, taaki geometric problems ko equations se solve kiya ja sake.
Point (−5, −3) kis quadrant mein hai?
Quadrant III kyunki dono x aur y coordinates negative hain (bottom-left region).
Agar kisi point ka x-coordinate 0 hai, toh woh kahan hoga?
y-axis par (y ki value ke hisaab se origin ke upar ya neeche ho sakta hai, lekin x=0 matlab yeh vertical axis par hai).
Quadrants clockwise ki jagah counterclockwise kyun jaate hain?
Yeh mathematics mein positive rotation direction ke convention se match karta hai, wahi jo trigonometry aur unit circle mein hota hai (anticlockwise = positive angle).
