11. Introduction to Three Dimensional Geometry
11.1 Introduction
Three-dimensional geometry extends 2D coordinate geometry to space, allowing precise location of points using three coordinates (x, y, z). This is essential for modeling real-world objects like aircraft trajectories or molecular structures.
11.2 Coordinate Axes and Planes
Coordinate Axes
- X-axis: Horizontal axis
- Y-axis: Vertical axis
- Z-axis: Depth axis
These axes are mutually perpendicular and intersect at the origin (0,0,0).
Coordinate Planes
- XY-plane: X-Y plane (z=0)
- YZ-plane: Y-Z plane (x=0)
- ZX-plane: Z-X plane (y=0)
These planes divide space into 8 octants.
11.3 Coordinates of a Point
A point P in 3D space is represented as (x, y, z) where:
| Coordinate | Distance from | Color Code |
|---|---|---|
| x | YZ-plane | Red |
| y | ZX-plane | Green |
| z | XY-plane | Blue |
11.4 Distance Between Two Points
For points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
Example
Distance between (1, 3, -2) and (2, 4, -1):
√[(2-1)² + (4-3)² + (-1+2)²] = √[1 + 1 + 1] = √3
Key Features Summary
| Concept | Visual Code | Key Formula |
|---|---|---|
| Axes | X, Y, Z | Mutually perpendicular |
| Planes | XY, YZ, ZX | Divide space into 8 octants |
| Distance | √[Δx² + Δy² + Δz²] | 3D extension of Pythagoras |