Introduction to 3D Geometry 🧊

2D se 3D — ek aur dimension add karo, duniya badal jaati hai! Z-axis ke saath teen coordinates se har point describe hota hai.

📦 Apna Kamra — 3D Space

Socho tum apne kamre mein khade ho. Ek kona dhundho jahan do deewarain aur zameen milti hain — yahi origin hai. Ek deewaar ke saath x-axis, doosri deewaar ke saath y-axis, aur chhath/zameen ki taraf z-axis. Ab kamre mein koi bhi point (x, y, z) — jaise ceiling par light ka position ya almaari ka corner — teen numbers se bilkul exactly describe ho sakta hai! Yahi hai 3D coordinate system.

1. 3D Coordinate System

2D mein sirf 2 axes the (x aur y) — humara plane. 3D mein teen mutually perpendicular axes hain: X-axis, Y-axis, Z-axis. Teeno ek common point par milte hain jise Origin O(0,0,0) kehte hain.

Har point ko ordered triple (x,y,z) se represent karte hain. Yahan x, y, z ko coordinates kehte hain — woh wo distances hain jo origin se us point tak teen axes ke parallel hain.

3D Coordinate Axes aur ek Point P(3,2,4)

O(0,0,0) X Y Z P(3,2,4) Right-hand rule: X=red, Y=green, Z=blue

2. Octants — 3D mein 8 Regions

2D plane mein 4 quadrants the — teen axes se space 8 parts mein bant jaata hai jise Octants kehte hain. Har octant mein x, y, z ka sign (+ ya −) alag hota hai.

Point ka octant dekhne ke liye bas uske x, y, z ke signs dekho!

I Octant

(+,+,+)

II Octant

(−,+,+)

III Octant

(−,−,+)

IV Octant

(+,−,+)

V Octant

(+,+,−)

VI Octant

(−,+,−)

VII Octant

(−,−,−)

VIII Octant

(+,−,−)

✅ Bolo yeh points kaunse octant mein hain

A(3, 2, 5) → (+,+,+) → I Octant
B(−1, 4, 2) → (−,+,+) → II Octant
C(2, −3, −1) → (+,−,−) → VIII Octant
D(−2, −1, 6) → (−,−,+) → III Octant

3. Coordinate Planes

Teeno axes se 3 planes bante hain jo do-do axes milne se banate hain:

Kisi point ka kisi plane par projection = us coordinate ko 0 kar do!

PlaneAxesFixed CoordExample Point on plane
XY-PlaneX & Yz = 0(3, 4, 0)
YZ-PlaneY & Zx = 0(0, 2, 5)
XZ-PlaneX & Zy = 0(1, 0, 7)

4. Distance Formula in 3D

2D mein Pythagoras theorem se distance formula tha: √((x₂−x₁)²+(y₂−y₁)²). 3D mein ek aur term add hota hai:

d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Intuition: Pehle xy-plane mein horizontal distance nikalo, phir z ke saath vertical — yahi 3D distance formula ka essence hai (Pythagoras do baar!).

✅ A(1,2,3) aur B(4,6,3) ke beech distance nikalo

d = √[(4−1)² + (6−2)² + (3−3)²]
= √[9 + 16 + 0] = √25 = 5

z-coordinates same hain → yeh points XY-plane ke parallel plane par hain → distance same as 2D!

✅ P(2,−1,2) ki origin se distance nikalo

d = √[2² + (−1)² + 2²] = √[4+1+4] = √9 = 3

5. Section Formula in 3D

Jab ek point P line segment AB ko m:n ratio mein divide kare (internally), toh P ki coordinates:

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))

Midpoint (m=n=1): P = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

✅ A(1,2,3) aur B(5,6,7) ka midpoint nikalo

Midpoint = ((1+5)/2, (2+6)/2, (3+7)/2)
= (3, 4, 5)

6. Important Points on Axes aur Planes

Kisi bhi point ka special location samajhna bahut important hai:

7. Practice Questions

Q1. A(2,3,−4) aur B(−1,2,2) ke beech distance nikalo.

Solution

d=√[(−3)²+(−1)²+(6)²]=√[9+1+36]=√46

Q2. (3,−4,5), (−2,1,0), (−1,2,4) — har ek kaunse octant mein hai?

Solution

(3,−4,5)→IV | (−2,1,0)→YZ plane (x<0,y>0,z=0) → II/VI boundary | (−1,2,4)→II Octant

Q3. A(1,1,1) aur B(7,7,7) ko 1:2 mein divide karne wala point nikalo.

Solution

P=((1×7+2×1)/3, (1×7+2×1)/3, (1×7+2×1)/3) = (9/3,9/3,9/3) = (3,3,3)

Q4. Prove karo ki A(0,1,2), B(2,−1,3), C(1,−3,1) ek isosceles triangle banate hain.

Solution

AB=√[4+4+1]=3, BC=√[1+4+4]=3, AC=√[1+16+1]=√18. AB=BC, AC≠AB → isosceles ✓

Q5. Origin se distance 7 wale points X-axis par kaunse hain?

Solution

X-axis par point (x,0,0). Distance=|x|=7 → points = (7,0,0) aur (−7,0,0)

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