Understanding the Equation of a Plane in Mathematics: Definition, Examples, and Solutions

plane

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely in all directions

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely in all directions. It has no thickness or curvature and is often represented as a horizontal surface that extends infinitely in the x and y directions.

A plane can be defined by a couple of key elements:

1. Point and normal vector: A plane can be uniquely defined by a specific point on the plane and a vector that is perpendicular to the plane, known as the normal vector. The normal vector determines the orientation and tilt of the plane.

2. Equation: Another way to define a plane is by using an equation. A general equation for a plane in 3D space is Ax + By + Cz + D = 0, where A, B, C are the coefficients of the x, y, and z variables respectively, and D is a constant term. This equation represents all points (x, y, z) that satisfy the relationship.

To better understand how to work with planes, let’s consider an example:

Example: Find the equation of a plane that contains the point (1, 2, 3) and is perpendicular to the vector V = (2, -1, 4).

To find the equation, we need a point on the plane and the normal vector to the plane. In this case, the point (1, 2, 3) is given, and the vector V = (2, -1, 4) is perpendicular to the plane.

The equation of the plane can be found using the point-normal form:

Ax + By + Cz + D = 0

We can substitute the coordinates of the given point (1, 2, 3) into the equation:

A(1) + B(2) + C(3) + D = 0

Simplifying, we get:

A + 2B + 3C + D = 0 —-(1)

Next, we need to find the values of A, B, C, and D using the normal vector V = (2, -1, 4).

Since V is perpendicular to the plane, the dot product of V with any vector on the plane should be zero.

The dot product of the normal vector (A, B, C) with the vector V = (2, -1, 4) is:

A(2) + B(-1) + C(4) = 0

2A – B + 4C = 0 —-(2)

Now we have a system of two equations (equations 1 and 2) with four variables (A, B, C, and D).

By solving this system of equations, we can find the specific values of the coefficients A, B, C, and D, which define the equation of the plane.

This could involve further steps such as substitution, elimination, or matrix operations, depending on the specific values involved. However, a detailed solution for this specific system is not provided as it would require more information about the problem.

In summary, a plane is a two-dimensional surface that extends infinitely in all directions. It can be defined by a point and a normal vector or by an equation. To find the equation of a plane, you need a point on the plane and information about its orientation, such as a normal vector.

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