Points, lines, & planes

Points, Lines, and Planes are fundamental concepts in geometry:

Point: A location in space with no dimensions—no length, width, or height. It is usually represented by a dot and labeled with a capital letter (e.g., point A).
Line: A straight, one-dimensional figure that extends infinitely in both directions. It has no thickness and is defined by at least two points. It can be represented with arrows on both ends (e.g., line AB).
Plane: A flat, two-dimensional surface that extends infinitely in all directions. It is defined by three non-collinear points or by a line and a point not on that line. Planes can be visualized as a sheet that has length and width but no thickness.

These concepts form the basis for understanding more complex geometric principles.

In this article

Part 1: Specifying planes in three dimensions

In a three-dimensional space, a plane can be defined by three points it contains, as long as those points are not on the same line. Learn more about it in this video.

When studying "Specifying planes in three dimensions," key points to focus on include:

Equation of a Plane: Understand the general form of a plane's equation in 3D space, typically expressed as $ax + by + cz + d = 0$ , where $(a, b, c)$ is the normal vector.
Normal Vectors: Recognize the significance of the normal vector, which is perpendicular to the plane and defines its orientation in space.
Point-Normal Form: Familiarize yourself with the point-normal form of a plane, expressed as $(x - x_0, y - y_0, z - z_0) \cdot (a, b, c) = 0$ , where $(x_0, y_0, z_0)$ is a point on the plane.
Intercept Form: Learn the intercept form of a plane's equation and how to derive it from the general equation.
Finding the Intersection of Planes: Study how to find the line of intersection between two planes.
Distance from a Point to a Plane: Understand the formula to calculate the perpendicular distance from a point to a given plane.
Graphical Representation: Develop the ability to visually represent planes in three-dimensional space for better conceptual understanding.
Parametric and Symmetric Forms: Explore how to express a plane in parametric and symmetric equations.

Concentrating on these points will solidify your understanding of how to specify and work with planes in three-dimensional geometry.

Points, lines, & planes

Part 1: Specifying planes in three dimensions

Related topics

Lines, line segments, and rays

Measuring segments

Parallel and perpendicular

Geometric definitions

The golden ratio