Section 12.1

The 3-D Coordinate System

Moving from to : Octants, Spheres, and Cylinders.

Introduction

In 2D calculus, we work in the -plane. In 3D calculus, we add a third axis, , perpendicular to the other two. This creates three "coordinate planes" () and divides space into eight "octants".

A key concept is how equations change meaning. is a circle in 2D, but in 3D it becomes a cylinder because is unrestricted.

Interactive: From Circle to Cylinder

The equation describes a surface (cylinder) because can be anything.

Key Formulas

Distance Formula in 3D

Extension of Pythagorean theorem:

Equation of a Sphere

Center and Radius :

Surfaces vs curves

1 variable fixed (e.g., ): A Plane.
1 variable missing (e.g., ): A Cylinder (surface extended along missing axis).
All variables present (e.g., ): A Surface (Plane, Sphere, etc).

Worked Examples

Example 1: Distance Between Points (Level 1)

Find the distance between and .

Step 1: Substitute into Formula

Step 2: Simplify

Example 2: Analyzing a Sphere (Level 2)

Find the center and radius of the sphere given by .

Strategy: Complete the square for each variable.

Group terms:
Complete squares:
Add and to both sides.
Factor:

Center: , Radius:

Example 3: Identifying Surfaces (Level 3)

Describe the graph of in .

In 2D (), this is a line with slope 2.
In 3D (), the variable is missing. This means can be any real number.

For every height , we have the same line. Stacked together, these lines form a Vertical Plane.