What Is a Plane in Euclidean Geometry? Core Terms and Real-Life Uses
Plane Definition
A plane is a flat, two-dimensional surface that can extend infinitely far. This means that there are no constraints in a plane. The examples of these planes can be seen in coordinate geometry and are very common in our world. The plane math definition or plane definition geometry is the same. Planes have no thickness or width, which makes it completely two dimensional. A mathematical plane can consist of a point, a line, or/and a three-dimensional space. Planes could also be subspaces of higher dimensional spaces, like the walls of a room, being extended infinitely or they can also be independently existing. This can be seen in Euclidean Geometry. Here you can understand the plane geometry definition and example.
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Important Terms Plane
Point:
Point is an element in any dimensional space. The whole of Euclidean Geometry is based on points. They are defined by axioms and are said to have no length, area, volume, or dimensional attributes.
Line:
Lines can be considered as a set of points that have no curvature and are straight objects. They have the only dimension as length. They can be infinite or can be bounded between 1 point (ray) or two points (line segment). Lines also do not have any predefined definition and are described using Euclidean Axioms.
2-Dimensional Space:
This is a geometric aspect where two values or parameters are required to find the position of a point, line, or shape. The two-dimensional space can be represented as R2. Normally, the two parameters in coordinate axes are taken as x and y parameters.
3-Dimensional Space:
This is a geometric aspect where three values or parameters are required to find the position of a point, line, plane, or object. The three-dimensional space can be represented as R3. Normally, the three parameters in coordinate axes are taken as x, y, and z parameters.
Axiom:
Axiom, postulates, or assumptions are statements that are taken as true without proof to use in other proofs. They are accepted without controversy or questioning since they are well known.
Euclidean Geometry:
This is a mathematical system constructed on the basis of dimensions and axioms. Euclid described these ideas in his textbook: the Elements.
Example of a Plane:
In our three-dimensional world, finding examples of planes is very hard. You can consider a sheet of paper with very negligible thickness as a plane. The surface of a table or a flat surface can be considered as a plane if they can be considered to have negligible thickness (which is almost impossible in the real world!)
Properties of Planes According to Euclidean Geometry
In any dimension planes are determined by the following:
Presence of three non-collinear points (points that do not lie on the same line)
A-line and a point that does not lie on that line.
Two lines that are distinct but intersect each other at a point.
Two lines that are distinct but parallel to each other.
Here are some properties of planes for specifically three-dimensional planes:
Two planes that are distinct can either be parallel or intersect each other in a line.
A line can either be parallel to a plane, can intersect it at a single point, or can be contained in the plane.
Two lines that are distinct and perpendicular to the same plane should be parallel to each other.
Two planes that are distinct and perpendicular to the same line should be parallel to each other.
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Planes can be found if there are different parameters given. For planes in three-dimensional spaces, they can be represented in many ways.
Point Normal Form:
If r0 is the position vector of a point P0 = (x0, y0, z0) and n=(a,b,c) is a non-zero vector, the plane can be determined.
The dot product of n and (r - r0) will be 0.
After expanding, this becomes:
a (x-x0) + b (y-y0) + c (z-z0) = 0
This form is the point-normal form of the plane. The same can be represented as a linear equation.
ax + by + cz + d = 0
where,
d = -(ax0 + by0 + cz0 )
Point with a plane and two vectors lying on it:
We can represent this as
r = r0 + SV + tw,
where ‘s’ and ‘t’ belong to real numbers and v and w are linearly independent vectors lying on the plane. r0 is the position vector of a fixed point on the plane.
Note: Plane and Plain are different things.
Geometrically, a plane is a flat surface with no width or thickness, whereas a plain is a flat expanse of land that is used in geographical terms to describe the terrain of a place. A plane can also be considered as an airplane.
FAQs on Planes in Euclidean Geometry: Definitions and Key Properties
1. What is a plane in Euclidean geometry?
In Euclidean geometry, a plane is defined as a flat, two-dimensional surface that extends infinitely in all directions. It has length and width but no thickness. It is one of the fundamental undefined concepts, along with points and lines, upon which geometry is built.
2. What are some common real-world examples that approximate a geometric plane?
While a true geometric plane is an abstract concept, several real-world objects can be used as examples. These include the surface of a calm lake, a flat tabletop, a whiteboard, or a sheet of paper. These are considered approximations because they have a physical thickness and do not extend infinitely, unlike a true mathematical plane.
3. What are the different conditions that can define a unique plane in 3D space?
A unique plane in three-dimensional space can be determined by any of the following conditions:
- Three non-collinear points (points that do not lie on the same straight line).
- A line and a point that is not on that line.
- Two distinct lines that intersect at a single point.
- Two distinct lines that are parallel to each other.
4. What are the primary ways to represent the equation of a plane in 3D?
The two primary forms for the equation of a plane are:
- Point-Normal Form: a(x - x₀) + b(y - y₀) + c(z - z₀) = 0, where (x₀, y₀, z₀) is a point on the plane and (a, b, c) are the components of a vector normal (perpendicular) to the plane.
- General Form: ax + by + cz + d = 0, which is a linear equation derived from the point-normal form.
5. How does a Euclidean plane differ from a Cartesian plane?
A Euclidean plane is the fundamental geometric space as described by Euclid's axioms, focusing on properties like points, lines, and angles without a fixed reference system. A Cartesian plane is a type of Euclidean plane that has been equipped with a coordinate system—typically the x and y-axes—which allows every point on the plane to be uniquely identified by an ordered pair of numbers (x, y).
6. Why are three non-collinear points required to define a unique plane, instead of just two?
Two points can only define a unique straight line. An infinite number of different planes can pass through that single line, much like the pages of a book can rotate around its spine. A third point that is not on that line (non-collinear) is needed to fix the plane's position and prevent it from 'rotating', thus defining one single, unique plane.
7. What is the significance of the normal vector in the equation of a plane?
The normal vector is crucial because it defines the orientation or 'tilt' of the plane in 3D space. It is a vector that is perpendicular (orthogonal) to every line and vector lying within the plane. The equation of a plane is fundamentally built on the principle that the dot product of the normal vector and any vector lying on the plane is zero, signifying their perpendicular relationship.
8. What is the difference between the geometric term 'plane' and the geographical term 'plain'?
Though they sound similar, 'plane' and 'plain' have very different meanings. A plane is a mathematical, abstract concept of a perfectly flat 2D surface with no thickness. A plain is a geographical term for a large, flat area of land with few trees. The confusion is understandable, as a geographical plain is a real-world example of a nearly flat surface.
