Lines and Planes in Stereometry

Stereometry is a branch of geometry that studies figures in three-dimensional space. The main elements of stereometry are points, lines, and planes.
Understanding their properties and mutual arrangement is key to studying complex spatial constructions.

Line in Space

A line is an infinite one-dimensional object, fully defined by two points. In space, a line can occupy various positions relative to other lines and planes.
Key properties and concepts:

Defining a Line: A line in space is defined by:

• Two points that belong to it.
• The equation of the line in parametric or canonical form.

Mutual Arrangement of Lines:

• Intersecting: Lines have one common point.
• Parallel: Lines do not intersect and lie in the same plane.
• Skew: Lines do not intersect and do not lie in the same plane.

Plane in Space

A plane is an infinite two-dimensional set of points. It can be defined in several ways:

• By three points that do not lie on the same line.
• By a line and a point not on that line.
• By two intersecting or parallel lines.

Equation of a Plane: A plane in space can be defined by the equation:

where A, B, and C are coefficients that determine the direction of the normal vector to the plane, and D is the constant term.

Mutual Arrangement of Lines and Planes

The mutual arrangement of a line and a plane can be as follows:

• The line lies in the plane: All points of the line belong to the plane.
• The line intersects the plane: The line has one common point with the plane.
• The line is parallel to the plane: The line does not intersect the plane.

For two planes, the possible arrangements are:

• Intersecting: Planes intersect along a line.
• Parallel: Planes do not have common points.
• Coinciding: Planes completely overlap each other.

Angles in Stereometry

Stereometry also studies angles between:

• Lines: The angle between intersecting lines is defined as the angle between their direction vectors.
• Line and Plane: This is the angle between the line and its orthogonal projection onto the plane.
• Planes: The angle between planes is equal to the angle between their normal vectors.

Fundamental Theorems

• Plane Axiom: Through three points that do not lie on the same line, there passes a unique plane.
• Parallelism Theorem: If two lines are parallel, then a plane passing through one of them will be parallel to the other.
• Intersection Theorem: If two planes intersect, their line of intersection is a line.

Additional Concepts

Vectors in Stereometry

Vectors are an important tool in stereometry for describing directions and magnitudes. A vector can be defined by the coordinates of its endpoints or through
its components.

Coordinate System

For convenience in working with spatial figures, a three-dimensional coordinate system is used, which allows for the unambiguous determination of the positions
of points in space.

Volumes and Surface Areas

Stereometry also studies the volumes and surface areas of three-dimensional figures, such as cubes, parallelepipeds, pyramids, cylinders, cones, and spheres. Formulas for calculating volumes and surface areas are important for practical applications in various fields.