Section 2.1 - Three-Dimensional Coordinate Systems

3D Space

In analytic geometry, three-dimensional space is described using three mutually perpendicular axes: the x-axis, y-axis, and z-axis. These axes intersect at the origin, denoted as O, and define the coordinate system for locating points in space.

• Coordinate Axes: The x, y, and z axes are perpendicular to each other and intersect at the origin.

• Coordinate Planes: The xy-plane contains the x- and y-axes; the yz-plane contains the y- and z-axes; the xz-plane contains the x- and z-axes.

• Octants: The three coordinate planes divide space into eight regions called octants.

• Point Representation: Any point P in space is represented by an ordered triple , where x, y, and z are the coordinates along the respective axes.

Example: The point P(3, -2, 5) is located 3 units along the x-axis, -2 units along the y-axis, and 5 units along the z-axis from the origin.

Distance Formula in 3D

The distance between two points and in three-dimensional space is given by:

• Application: Used to find the straight-line distance between any two points in 3D space.

Example: Find the distance from to .

Equations of Planes Parallel to Coordinate Planes

Planes in 3D space can be parallel to one of the coordinate planes. Their equations are simple forms:

• Parallel to xy-plane: The plane through parallel to the xy-plane is .

• Parallel to xz-plane: The plane through parallel to the xz-plane is .

• Parallel to yz-plane: The plane through parallel to the yz-plane is .

Example: Write the equation of the plane passing through that is parallel to the yz-plane. The equation is .

Equation of a Sphere

A sphere in 3D space is defined as the set of all points that are a fixed distance (radius) from a given center . The equation of a sphere is:

• Center:

• Radius:

• Special Case: If the center is the origin , the equation simplifies to .

Example: Find the equation of a sphere with radius 2 and center :

Example: Show that is the equation of a sphere, and find its center and radius.

Summary Table: Equations of Planes Parallel to Coordinate Planes

Additional info:

• In 3D analytic geometry, equations of planes and spheres are foundational for understanding surfaces and volumes.

• These concepts are essential for later topics such as vectors, lines, and calculus in multiple dimensions.