Exploring the Equation of a Plane Parallel to a Given Plane and Passing Through Specific Points

In this article, we delve into the mathematical concepts of planes and vectors, specifically focusing on finding the equation of a plane that is parallel to a given plane and passes through specified points. The problem we will address involves the vectors and equations related to the plane and how they can be used to derive the final equation. Let's break down the problem step-by-step to understand the logic and reasoning behind the solution.

Understanding the Given Problem

We are given two points, -1, -1, 6 and 2, 1, 1, and a plane equation 2x y z 4. Our goal is to find the equation of a plane that is parallel to the given plane and passes through the specified points. The key to solving this problem is understanding that parallel planes have the same normal vector.

Step 1: Identify the Normal Vector of the Given Plane

The normal vector of the plane 2x y z 4 is (2, 1, 1). This vector is perpendicular to the plane and defines its orientation.

Step 2: Verify if the Points Are on the Given Plane

To check if the points -1, -1, 6 and 2, 1, 1 lie on the plane, we substitute them into the plane equation.

For the point -1, -1, 6: 2(-1) (-1) 6 -2 - 1 6 3 ≠ 4 For the point 2, 1, 1: 2(2) 1 1 4 1 1 6 ≠ 4

Since the points do not satisfy the plane equation, they do not lie on the given plane. However, we are still tasked with finding a plane parallel to the given one that passes through these points.

Step 3: Find the Equation of the Required Plane

Given that the required plane is parallel to the given plane, it will have the same normal vector (2, 1, 1). Therefore, its equation will be of the form:

2x y z d

Step 4: Determine the Constant d

To find d, we use one of the given points. Let's use the point -1, -1, 6.

2(-1) (-1) 6 d

2(-1) (-1) 6 -2 - 1 6 3 d

Therefore, the equation of the plane is:

2x y z 3

We can verify this by substituting the other point 2, 1, 1 into the plane equation:

2(2) 1 1 4 1 1 6 ≠ 3

Since the second point does not satisfy the equation, it is not on the plane. Hence, no single plane can pass through both given points and be parallel to the given plane.

Conclusion

Using the given points and the properties of parallel planes, we derived the equation of a plane parallel to the given plane that passes through one of the points but not the other. This problem highlights the importance of vector operations and the use of plane equations in solving geometric problems.

Related Keywords

Plane equation, Parallel planes, Vector perpendicular