Coordinate system in 3D — x, y, z axes, octants

3.6.1 · Maths › 3D Geometry

1. Teen axes kya hote hain?

Ordered kyun? Kyunki $(2, 3, 5)$ aur $(5, 3, 2)$ alag-alag points hain. Triple mein position batati hai ki kaunsa ruler use ho raha hai.

Hum ise kaise draw karte hain (right-handed system): Apne right hand ki ungliyan $+ x$ ki taraf point karo, unhe $+ y$ ki taraf curl karo; tumhara thumb $+ z$ ki taraf point karega. Mathematics har jagah (cross products, etc.) yahi convention use karta hai.

Figure — Coordinate system in 3D — x, y, z axes, octants

2. Coordinate planes

3. Octants — space ko 8 boxes mein kaatna

Octant	$x$	$y$	$z$	Example point
I	+	+	+	$(2, 3, 4)$
II	−	+	+	$(- 2, 3, 4)$
III	−	−	+	$(- 2, - 3, 4)$
IV	+	−	+	$(2, - 3, 4)$
V	+	+	−	$(2, 3, - 4)$
VI	−	+	−	$(- 2, 3, - 4)$
VII	−	−	−	$(- 2, - 3, - 4)$
VIII	+	−	−	$(2, - 3, - 4)$

4. Special positions (signs ke liye intuition banao)

5. Origin se point ki distance (derive karo!)

Scratch se derivation. Lo $P (x, y, z)$ . Ise xy-plane par drop karo taaki foot $M (x, y, 0)$ mile.

Step 1 — floor mein distance: $O M = x^{2} + y^{2} (Kyun? xy-plane mein Pythagoras.)$

Step 2 — seedha upar jao: $P M = ∣ z ∣$ , aur $P M ⊥ O M$ (vertical edge floor ke perpendicular hai).

Step 3 — Pythagoras phir se triangle $O M P$ mein: $O P^{2} = O M^{2} + P M^{2} = (x^{2} + y^{2}) + z^{2} .$

6. Common mistakes (Steel-manned)

7. Feynman + memory aids

Recall 12-saal ke bacche ko explain karo (reveal karne ke liye click karo)

Apne bedroom ke corner ki imagine karo. Floor aur do walls us corner par milti hain — woh hai "zero, zero, zero". Kisi dost ko batane ke liye ki balloon kahan uda raha hai, tum kaho: "is wall ke along 2 metre jao, us wall ke along 3 metre, aur 4 metre upar." Woh teen numbers $(2, 3, 4)$ balloon ka address hain. Walls aur floor poore room ko (aur us se bahar ki space ko bhi) 8 boxes mein kaatti hain — woh hain octants, jaise 8 invisible rooms ek corner par milte hain.

8. Active recall — Flashcards

3D space mein kitne octants hote hain, aur kyun?

8, kyunki teen perpendicular planes space ko kaatti hain aur x, y, z mein se har ek + ya − ho sakta hai:

2^{3} = 8

Jis point ka exactly ek coordinate zero ho woh kahan hota hai?

Ek coordinate plane mein.

Jis point ke exactly do coordinates zero hon woh kahan hota hai?

Ek coordinate axis par.

xy-plane ke har point mein kaunsa coordinate zero hota hai?

z = 0

yz-plane mein kaunsa coordinate zero hota hai?

x = 0

Octant I ka sign pattern?

(+, +, +)

Octant VII ka sign pattern?

(-, -, -)

P (x, y, z)

ki origin se distance?

x^{2} + y^{2} + z^{2}

(2, 3, 6)

ki origin se distance?

7

Kya

(2, 3, 5)

aur

(5, 3, 2)

same point hain?

Nahi — triple ordered hota hai, isliye woh alag hain.

(3, - 2, 6)

kis octant mein hai?

Octant IV (signs

+, -, +

Axis orientation kaunsa hand-rule fix karta hai?

Right-hand rule (right-handed coordinate system).

Connections

Distance Formula in 3D — Section 5 ko kisi bhi do points tak generalise karta hai.
Section Formula in 3D — ek segment ko divide karna in coordinates ka use karta hai.
Direction Cosines and Direction Ratios — axes/lines ki direction describe karte hain.
Quadrants in 2D Coordinate Geometry — octants ka 2D purvaj.
Vectors in 3D — position vector $O P = x \hat{i} + y \hat{j} + z \hat{k}$ in axes par bana hai.
Equation of a Plane — coordinate planes sabse simple planes hain.