Definition properties and equation of a plane with examples
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 Explained
1. What is a plane in Euclidean geometry?
A plane in Euclidean geometry is a flat, two-dimensional surface that extends infinitely in all directions. It has no thickness and is made up of infinitely many points and lines.
- A plane is determined by three non-collinear points.
- It can also be determined by a line and a point not on the line.
- In coordinate geometry, a plane is often represented by a linear equation in three variables.
2. What is the equation of a plane in 3D geometry?
The general equation of a plane in 3D geometry is Ax + By + Cz + D = 0, where A, B, C, and D are constants. Here:
- (A, B, C) is the normal vector to the plane.
- x, y, and z are the coordinates of any point on the plane.
3. How do you find the equation of a plane given three points?
To find the equation of a plane through three non-collinear points, compute a normal vector using the cross product of two direction vectors and substitute into the plane formula.
- Step 1: Form two vectors from the three points.
- Step 2: Find their cross product to get the normal vector.
- Step 3: Use point-normal form: A(x − x₁) + B(y − y₁) + C(z − z₁) = 0.
4. What is the normal vector of a plane?
A normal vector of a plane is a vector that is perpendicular to the plane. In the plane equation Ax + By + Cz + D = 0, the vector (A, B, C) is the normal vector.
- It determines the orientation of the plane.
- It is used to check perpendicularity between planes and lines.
- Two planes are parallel if their normal vectors are proportional.
5. How do you find the distance from a point to a plane?
The distance from a point (x₁, y₁, z₁) to a plane Ax + By + Cz + D = 0 is given by |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²). This formula measures the shortest (perpendicular) distance.
- Substitute the point coordinates into the numerator.
- Compute the magnitude of the normal vector in the denominator.
6. What is the difference between a line and a plane?
The main difference is that a line is one-dimensional, while a plane is two-dimensional.
- A line extends infinitely in one direction (length only).
- A plane extends infinitely in two directions (length and width).
- A line can lie in a plane or intersect it at a single point.
7. When are two planes parallel?
Two planes are parallel if their normal vectors are proportional, meaning A₁/A₂ = B₁/B₂ = C₁/C₂.
- They have the same orientation.
- They never intersect.
- Their equations differ only in the constant term.
8. How do you find the angle between two planes?
The angle between two planes is the angle between their normal vectors. It is calculated using cosθ = (n₁ · n₂) / (|n₁||n₂|).
- Find the normal vectors of both planes.
- Compute their dot product.
- Divide by the product of their magnitudes.
9. What is the intercept form of a plane?
The intercept form of a plane is x/a + y/b + z/c = 1, where a, b, and c are the intercepts on the coordinate axes.
- a is the x-intercept.
- b is the y-intercept.
- c is the z-intercept.
10. How many points are needed to determine a plane?
A plane is uniquely determined by three non-collinear points.
- If the three points lie on the same line, infinitely many planes can pass through them.
- If they are not collinear, exactly one plane passes through all three.
