What is a plane: definition, properties, equations

In this publication, we will consider the definition of a plane, list its main properties, and also give various types of equations with which it can be specified.

Plane definition and properties

Plane is a surface that completely contains every line connecting any of its points.

Plane properties:

• Two planes are either parallel or intersect in a straight line.
• A straight line is either parallel to the plane, or is on it, or intersects it at one point.
• If two lines are perpendicular to the same plane, then they are parallel.
• Parallel planes are perpendicular to the same line.

Plane equations

General view

The plane can be given by a first degree equation:

Ax + By + Cz + D = 0

Note: A, B и C can be equal to zero, and at the same time.

Equation in segments

Suppose there is a plane that intersects the coordinate axes Ox, Oy и Oz at points with coordinates (a, 0, 0), (0, b, 0) и (0, 0, c). In this case, its equation is:

Plane passing through a point perpendicular to the normal vector

If we know the coordinates of a point in the plane, for example, M (x₀y₀, of₀), as well as the normal vector of the plane, for example, n = {A; B; C}, then the equation can be given by the formula:

A(x – x₀) + B(and – and₀) + C(z – z₀) = 0

A plane passing through three points that do not lie on the same line

If the coordinates of three points belonging to the same plane are known, for example, A (x₁y₁, of₁), B (x₂y₂, of₂) и C (x₃y₃, of₃), then its equation will be: