Three-Dimensional Coordinate Systems

Introduction to Three-Dimensional Coordinates

The three-dimensional rectangular coordinate system is fundamental in multivariable calculus and analytic geometry. It extends the two-dimensional Cartesian plane by introducing a third axis, allowing for the representation of points, lines, and surfaces in space.

• Three Perpendicular Axes: The system consists of the x-axis, y-axis, and z-axis, all perpendicular to each other and intersecting at the origin (0, 0, 0).

• Notation: Each axis is a number line representing all real numbers in R. The three-dimensional system is denoted by R3.

• Right-Hand Rule: The orientation of the axes follows the right-hand rule, which is standard in mathematics and physics for determining the positive direction of the axes.

Example: The point (2, -1, 3) is located 2 units along the x-axis, -1 unit along the y-axis, and 3 units along the z-axis from the origin.

Coordinate Planes

Pairs of coordinate axes define the three principal coordinate planes in R3. These planes are essential for visualizing and describing geometric objects in space.

• xy-plane: Defined by z = 0 (all points where the z-coordinate is zero).

• yz-plane: Defined by x = 0 (all points where the x-coordinate is zero).

• xz-plane: Defined by y = 0 (all points where the y-coordinate is zero).

• Parallel Planes: The equations x = a, y = b, and z = c describe planes parallel to one of the coordinate planes, where a, b, and c are constants.

Example: The plane x = 2 is parallel to the yz-plane and consists of all points where the x-coordinate is 2.

Distance Formula in Three Dimensions

The distance between two points in three-dimensional space can be found using an extension of the Pythagorean theorem.

• Distance Formula: For points P = (x0, y0, z0) and Q = (x1, y1, z1):

• Application: This formula is used to compute the straight-line (Euclidean) distance between any two points in space.

Example: The distance between P = (1, 2, 3) and Q = (4, 6, 3) is:

Equation of a Sphere

The standard equation for a sphere in three-dimensional space is derived from the distance formula. A sphere is the set of all points at a fixed distance (radius) from a given center.

• Equation of a Sphere: For a sphere with center (x0, y0, z0) and radius r:

Example: The equation of a sphere with center (2, -1, 3) and radius 4 is:

Additional info: The right-hand rule is used to determine the orientation of the axes in three-dimensional space. The coordinate planes are fundamental for describing regions and surfaces in multivariable calculus. The distance formula and the equation of a sphere are foundational for later topics such as vectors, surfaces, and integration in space.