Three-Dimensional Coordinate Systems

Definition

Analytic geometry is extended to three dimensions by adding a z-axis perpendicular to both the x- and y-axes. This creates a system where any point $P$ is uniquely located by an ordered triple $(x, y, z)$ .

How to read: “P at ordered triple x, y, z.”
Meaning: Three mutually perpendicular axes locate any point in space with three numbers.

Why It Matters

The transition from 2D to 3D is the transition from a map to the world. 3D coordinate systems are the foundational scaffolding for all physical engineering, robotics, and aerospace, allowing us to precisely locate and manipulate objects in the real, volumetric universe.

Core Concepts

Right-Handed System: The standard orientation where the axes follow the right-hand rule.
Coordinate Planes:
- $xy$ $x y$ -plane ( $z=0$ $z = 0$ ).
  - How to read: “x-y plane, where z equals zero.”
  - Meaning: The horizontal floor/ceiling plane at height zero.
- $yz$ $y z$ -plane ( $x=0$ $x = 0$ ).
  - How to read: “y-z plane, where x equals zero.”
  - Meaning: The side wall plane at $x = 0$ .
- $xz$ $x z$ -plane ( $y=0$ $y = 0$ ).
  - How to read: “x-z plane, where y equals zero.”
  - Meaning: The front/back wall plane at $y = 0$ .
Midpoint Formula: For points $P_1(x_1, y_1, z_1)$ $P_{1} (x_{1}, y_{1}, z_{1})$ and $P_2(x_2, y_2, z_2)$ $P_{2} (x_{2}, y_{2}, z_{2})$ , the midpoint is $M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$ $M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$ .
- How to read: “The midpoint M has coordinates x one plus x two all over two, y one plus y two all over two, and z one plus z two all over two.”
- Meaning: The center point halfway between two points in 3D space — each coordinate is the arithmetic mean of the corresponding endpoints.
Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$ .
- How to read: “The distance d equals the square root of the quantity x two minus x one squared, plus the quantity y two minus y one squared, plus the quantity z two minus z one squared.”
- Meaning: 3D Pythagorean theorem—straight-line distance between two points.

Three-Dimensional Coordinate Systems

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes