Mathematics · Ch 5 — Introduction to Three-Dimensional Geometry
Introduction
Introduction
Moving from Two Dimensions to Three
You already know how to fix a point's position on a flat surface like a sheet of paper. You draw two perpendicular lines — the x‑axis and the y‑axis — and then describe the point by two numbers: its perpendicular distances from these axes. Those two numbers are its coordinates.
But the world around you is not flat. A ball thrown in the air, an aeroplane flying from Delhi to Mumbai, the tip of a fan hanging from the ceiling — none of these lie in a single plane. To locate such a point in space, two numbers are not enough.
Think of a room. Suppose you want to describe the position of the lowest tip of an electric bulb hanging from the ceiling. You need to know:
- how far it is from one wall (say the wall on your left),
- how far it is from the adjacent wall (the wall in front of you),
- and how high it is above the floor.
These three perpendicular distances — measured from three mutually perpendicular planes — are the three coordinates of the point in space. So, a point in space has three coordinates, not two.
In this chapter, you will learn the basic geometry of three‑dimensional space — how to set up the coordinate system, how to find distances between points, and how to work with coordinates in three dimensions.
The three planes you use are like the floor and two adjacent walls of a room. They are mutually perpendicular — each plane is at right angles to the other two. This is the natural extension of the two‑axis system you already know.
The Core Idea
| Two‑Dimensional Geometry | Three‑Dimensional Geometry |
|---|---|
| Two perpendicular axes (x and y) | Three mutually perpendicular planes |
| Two coordinates (x, y) | Three coordinates (x, y, z) |
| Locates points in a plane | Locates points in space |
The jump from 2D to 3D is straightforward: you add one more perpendicular direction. The three numbers that locate a point are called its coordinates with respect to the three coordinate planes.
A point in space requires three numbers — its perpendicular distances from three mutually perpendicular planes. These three numbers are its coordinates.