Find Equation Of Plane Through Point And Parallel To Plane

Finding the Equation of a Plane Through a Point and Parallel to Another Plane

Finding the equation of a plane that passes through a given point and is parallel to another given plane is a common problem in three-dimensional geometry. This process relies on understanding the relationship between the normal vector of a plane and its equation. This article will guide you through the steps, providing clear explanations and examples.

Meta Description: Learn how to determine the equation of a plane that passes through a specific point and remains parallel to a given plane. This guide provides a step-by-step approach with clear examples.

Understanding Plane Equations

The equation of a plane is typically written in the form:

Ax + By + Cz = D

where A, B, and C are the components of the normal vector (a vector perpendicular to the plane), and D is a constant. The normal vector is crucial because parallel planes share the same normal vector.

Step-by-Step Process

Here's how to find the equation of a plane given a point and a parallel plane:

1. Identify the Normal Vector: The first step is to determine the normal vector of the given plane. If the equation of the given plane is already in the form Ax + By + Cz = D, then the normal vector is simply <A, B, C>.

2. Use the Same Normal Vector: Since parallel planes share the same normal vector, the plane you're looking for will have the same normal vector as the given plane. Therefore, you already know the A, B, and C values for your new plane's equation.

3. Substitute the Point: You are given a point (x₀, y₀, z₀) that lies on the plane you want to find. Substitute the coordinates of this point into the general equation of a plane (Ax + By + Cz = D) along with the A, B, and C values obtained from the normal vector. This allows you to solve for D.

4. Write the Equation: Once you have solved for D, substitute this value back into the general plane equation along with the A, B, and C values from the normal vector. This is the equation of the plane that passes through the given point and is parallel to the given plane.

Example

Let's say we want to find the equation of the plane that passes through the point (2, 1, 3) and is parallel to the plane 2x - y + 3z = 5.

1. Normal Vector: The normal vector of the plane 2x - y + 3z = 5 is <2, -1, 3>.

2. Same Normal Vector: Our new plane will also have the normal vector <2, -1, 3>.

3. Substitute the Point: Substitute the point (2, 1, 3) and the normal vector into the equation:

   2(2) - 1(1) + 3(3) = D

   4 - 1 + 9 = D

   D = 12

4. Equation of the Plane: The equation of the plane is therefore:

   2x - y + 3z = 12

Handling Different Plane Equation Forms

Sometimes, the equation of the given plane might not be in the standard form. For example, you might encounter a plane described parametrically or using a different point-normal form. In such cases, you first need to convert the given plane's equation into the standard form (Ax + By + Cz = D) to easily extract the normal vector.

Conclusion

Finding the equation of a plane parallel to a given plane and passing through a given point is a straightforward process once you understand the significance of the normal vector. By following the steps outlined above, you can confidently solve this type of problem in three-dimensional geometry. Remember to always check your work by verifying that the given point satisfies the equation of the plane you derived.