What is a Plane?

View Notes

What is a Plane?

In geometry, a plane prolongs itself indefinitely in two dimensions. It doesn't have a width. In coordinate geometry, we can see an instance of a plane. In a plane, the coordinates determine the location of points.

A plane is a flat, two-dimensional surface in mathematics that stretches infinitely far. A plane is a geometry of two dimensions that may consist of a point, a line and a space of three dimensions. Planes may appear as subspaces of a few higher-dimensional spaces, such as extremely distant walls of space, or these walls undergo an autonomous life on their own, as in the Euclidean geometry context.

A plane with no thickness is a flat surface.

Plane

There are three dimensions to our universe, but on a plane, there are only two dimensions:

Scale and length make a plane

x and y construct a plane as well. There is no thickness in a plane, and it goes on forever.

As you read the description above, it is not possible that such a thing actually exists. Imagine a sheet of metal that is smooth. In both directions, now make it infinitely wide. This implies that you never hit its edges, no matter how far you go. Now imagine that it's so thin that there's no thickness at all. It remains, despite this, absolutely rigid and smooth. In geometry, this is the 'plane.'

Drawing planes is complicated because the edges have to be drawn. When you see an image of a plane, always note that it does not actually have edges and that it is infinitely big. There are two dimensions of the plane: length and width. However, as the plane is infinitely wide, it is not possible to calculate the length and width.

Just as two points define a line, a plane is defined by three points. There is only one plane that includes all three, given the three points that are not collinear.

What's the Point?

A point is a position with no distance, i.e. no width, no length and no depth in a plane.

What's that Line?

A line is a series of points in opposite directions that extend indefinitely. There is only one dimension to it, i.e. length. Collinear points are called the points which lie on the same side.

Plane in Algebra

The points are plotted in the coordinate plane in algebra, which denotes an instance of a geometric plane. There is a number line in the coordinate plane, running indefinitely left to right, and another one extending infinitely up and down. The full teamwork plane can not be shown.

Arrows at the ends of the number lines reflect the fact that it stretches indefinitely along the x-axis and the y-axis. Where the plane stretches indefinitely, these number lines are two-dimensional. The point or a line plotted does not have any thickness when we plot the graph in a plane.

Intersecting Planes

In three-dimensional space, two planes can be connected in three ways -

• They can be parallel to each other

• They can be identical

• They can intersect each other

The method of obtaining the equation of the intersection line connecting two planes is to determine the set of points that satisfy both planes' equations. The plane equation consists of three variables and there are two equations (because of two planes). Hence, a relationship between the three variables, which is equal to the equation of the intersection line, will be given by solving the concurrent equations.

The method of obtaining the equation of the intersection line connecting two planes is to determine the set of points that satisfy both planes' equations. Since the plane equation consists of three variables and there are two equations (because of two planes), a relationship between the three variables, which is equal to the equation of the intersection line, will be given by solving the concurrent equations.

Properties of a Plane

• If there are two separate planes in a three-dimensional space, they are either parallel to each other or intersect in a line.

• A line may be parallel to a plane, at a single point intersecting the plane or is present in the plane.

• If two distinct lines are perpendicular to the same plane, then they have to be parallel to each other.

• If two distinct planes are perpendicular to the same line, they must be parallel to each other.

Fun Fact

Plane versus Simple

A flat surface with no thickness is a' plane' in geometry.

However, "plain" is a large flat expanse of land. It is flat as well, but not in the pure sense of geometry that we use. There are also other meanings of both words.

A plane may also refer to an aircraft, a stage, or a tool to cut flat stuff.

Simple can mean without special objects or well understood,

Points are centred on the plane using their coordinates in another branch of mathematics called coordinate geometry - two numbers that indicate where the point is situated. The plane is assumed to have two scales at right angles to do this. Any point on the plane can be uniquely represented using a pair of numbers.

Q1. Define a Plane?

Ans: A plane is a flat, two-dimensional surface in mathematics that stretches infinitely far. A plane is a geometry of two dimensions that may consist of a point, a line and a space of three dimensions. Planes may appear as subspaces of a few higher-dimensional spaces, such as extremely distant walls of space, or these walls undergo an autonomous life on their own, as in the Euclidean geometry context.

Q2. Explain the Properties of a Plane?

Ans: A plane is a flat, two-dimensional surface in mathematics that stretches infinitely far. A plane is a geometry of two dimensions that may consist of a point, a line and a space of three dimensions. Properties of a Plane:

• If there are two separate planes in a three-dimensional space, they are either parallel to each other or intersect in a line,

• A line may be parallel to a plane, at a single point intersecting the plane or is present in the plane.

• If two distinct lines are perpendicular to the same plane, then they have to be parallel to each other.

• If two distinct planes are perpendicular to the same line, they must be parallel to each other.