📐 Three Dimensional Geometry (Class 12 NCERT) – Summary
📌 1. Introduction
Three Dimensional Geometry deals with the study of points, lines, and planes in space 🌌. It extends two-dimensional geometry by introducing a third axis.
📍 2. Coordinate System in Space
A point in space is represented by an ordered triplet (x, y, z).
- ➡️ x-coordinate: distance along x-axis
- ⬆️ y-coordinate: distance along y-axis
- 🔼 z-coordinate: distance along z-axis
📏 3. Distance Between Two Points
The distance between A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is:
AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
✂️ 4. Section Formula
If point P divides the line joining A and B internally in the ratio m:n:
P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))
🧭 5. Direction Cosines & Direction Ratios
Direction cosines (l, m, n) represent the orientation of a line in space.
- ✅ l² + m² + n² = 1
- 📐 Direction ratios are proportional to direction cosines
📈 6. Equation of a Line
🧮 a) Vector Form
r = a + λb
📊 b) Cartesian Form
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
🔄 7. Angle Between Two Lines
The angle θ between two lines is given by:
cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / √[(a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²)]
🟦 8. Plane
📄 a) General Equation
ax + by + cz + d = 0
📐 b) Normal Form
x cosα + y cosβ + z cosγ = p
📍 9. Distance of a Point from a Plane
Distance of (x₁, y₁, z₁) from the plane ax + by + cz + d = 0:
|ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)
🧩 10. Angle Between Two Planes
cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / √[(a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²)]
📐 11. Shortest Distance Between Two Skew Lines
The shortest distance is the length of the perpendicular between two non-intersecting lines 🚀.
It is calculated using vector cross products.
🎯 12. Key Applications
- 📦 Finding distances and angles in space
- 🧠 Understanding relative positions of lines and planes
- ⚙️ Applications in physics, engineering, and graphics
✅ Conclusion
Three Dimensional Geometry is a vital chapter that builds strong spatial visualization skills and lays the foundation for advanced mathematics and real-world applications 🌟.