Vectors and the Geometry of Space – Study Notes (Thomas’ Calculus, Ch. 12.1)

Vectors and the Geometry of Space

Three-Dimensional Coordinate Systems

Three-dimensional coordinate systems extend the familiar two-dimensional Cartesian plane into space, allowing us to describe points, lines, and planes in three dimensions. This section introduces the basic structure and interpretation of the 3D coordinate system.

• Definition: The three-dimensional Cartesian coordinate system consists of three perpendicular axes: the x-axis, y-axis, and z-axis. The point where all three axes intersect is called the origin, denoted as (0, 0, 0).

• Right-Handed System: The standard 3D Cartesian system is right-handed. If you point your right thumb along the positive x-axis and your index finger along the positive y-axis, your middle finger points in the direction of the positive z-axis.

• Coordinate Planes:

  • xy-plane: Defined by z = 0.

  • yz-plane: Defined by x = 0.

  • xz-plane: Defined by y = 0.

• Octants: The planes x = 0, y = 0, and z = 0 divide space into eight regions called octants. The first octant is where x, y, and z are all positive.

• Planes and Lines: Planes such as x = 2, y = 3, and z = 5 are parallel to the coordinate planes and pass through the point (2, 3, 5). Lines of intersection between these planes can be described by fixing two coordinates and varying the third.

Geometric Interpretation of Equations and Inequalities

Equations and inequalities in three variables can be interpreted as geometric objects in space.

• z ≥ 0: Represents the half-space consisting of all points on and above the xy-plane.

• x = -3: Represents the plane perpendicular to the x-axis at x = -3. This plane is parallel to the yz-plane and is located 3 units behind it.

• y = 0, x ≤ 0, y ≥ 0: Represents the second quadrant of the xy-plane.

Examples and Applications

• Example 1: The intersection of the planes x = 2, y = 3, and z = 5 is the point (2, 3, 5).

• Example 2: The region defined by z ≥ 0 is the set of all points above the xy-plane, including the plane itself.

• Example 3: The plane x = -3 is parallel to the yz-plane and passes through all points where x = -3.

Summary Table: Coordinate Planes and Their Equations

Plane	Equation	Description
xy-plane	z = 0	All points where z = 0
yz-plane	x = 0	All points where x = 0
xz-plane	y = 0	All points where y = 0

Key Formulas

• Distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Additional info:

• Understanding the geometric meaning of equations in three variables is foundational for studying vectors, planes, and surfaces in multivariable calculus.