Introduction to Three Dimensional Geometry
Ab tak humne points ko plane (2D) me \((x, y)\) se dikhaya. Real duniya 3D hai, isliye ab ek teesra axis \(z\) jodte hain. Is chapter me 3D me points, unke beech distance, aur section formula dekhenge.
1. Coordinate Axes aur Planes
3D me teen mutually perpendicular axes hote hain — \(x\), \(y\), \(z\) — jo origin \(O\) par milte hain. Inse teen coordinate planes bante hain: \(XY\), \(YZ\), aur \(ZX\) plane.
In planes se space 8 octants me bant jaata hai. Kisi point ko \((x, y, z)\) se likhte hain.
2. Coordinates of a Point
- \(x\) = \(YZ\)-plane se doori, \(y\) = \(ZX\)-plane se, \(z\) = \(XY\)-plane se.
- Origin: \((0, 0, 0)\).
- \(x\)-axis par koi point: \((x, 0, 0)\); \(XY\)-plane par: \((x, y, 0)\).
3. Distance Between Two Points
Points \(P(x_1, y_1, z_1)\) aur \(Q(x_2, y_2, z_2)\) ke beech doori: \[ PQ = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2} + (z_2 - z_1)^{2}}. \]
(Ye 2D distance formula ka hi 3D version hai — bas ek aur term \(z\) ka jud gaya.)
\(P(1, 2, 3)\) aur \(Q(4, 6, 3)\) ke beech doori: \[ PQ = \sqrt{(4-1)^{2} + (6-2)^{2} + (3-3)^{2}} = \sqrt{9 + 16 + 0} = 5. \]
4. Section Formula
Point \(R\) jo \(P(x_1, y_1, z_1)\) aur \(Q(x_2, y_2, z_2)\) ko ratio \(m : n\) me internally divide karta hai:
\[ R = \left( \frac{m x_2 + n x_1}{m + n},\ \frac{m y_2 + n y_1}{m + n},\ \frac{m z_2 + n z_1}{m + n} \right). \]
Midpoint
Ratio \(1 : 1\) lene par midpoint:
\[ M = \left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2},\ \frac{z_1 + z_2}{2} \right). \]
\(P(2, 4, 6)\) aur \(Q(6, 8, 10)\) ka midpoint: \[ M = \left( \frac{2+6}{2},\ \frac{4+8}{2},\ \frac{6+10}{2} \right) = (4, 6, 8). \]
Key Takeaways
- 3D me point \((x, y, z)\); teen axes space ko 8 octants me baantte hain.
- Distance: \(\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}+(z_2-z_1)^{2}}\).
- Section formula (\(m:n\) internal) aur midpoint formula yaad rakho.