Find the equation of plane which bisects the join of (x1,y1,z1) and (X2,y2,z2) perpendicularly

To find the equation of the plane that bisects the line segment joining the points (x1, y1, z1) and (x2, y2, z2) perpendicularly, we need to follow a systematic approach. The key steps involve determining the midpoint of the line segment and using the direction vector of the segment to identify the normal vector of the plane.

Step 1: Determine the Midpoint

The first thing we need to do is find the midpoint of the line segment connecting the two points. The midpoint M can be calculated using the formula:

M = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

This point M will lie on the bisecting plane.

Step 2: Find the Direction Vector

Next, we need to find the direction vector of the line segment, which can be obtained by subtracting the coordinates of the two points:

Direction vector, D = (x2 - x1, y2 - y1, z2 - z1)

Step 3: Establish the Normal Vector of the Plane

Since the plane we are looking for is perpendicular to the line segment, the direction vector D serves as the normal vector (N) of the plane:

N = (x2 - x1, y2 - y1, z2 - z1)

Step 4: Write the Equation of the Plane

The general form of the equation of a plane can be expressed as:

A(x - x0) + B(y - y0) + C(z - z0) = 0

Here, (A, B, C) are the components of the normal vector (N), and (x0, y0, z0) is a point on the plane, which in our case is the midpoint M.

Substituting these values into the equation, we get:

(x2 - x1)(x - (x1 + x2)/2) + (y2 - y1)(y - (y1 + y2)/2) + (z2 - z1)(z - (z1 + z2)/2) = 0

Step 5: Simplifying the Equation

To make this equation more understandable, we can expand and rearrange it as needed. This will yield the final equation of the plane that bisects the line segment perpendicularly.

Example

Let’s take an example to clarify this process. Suppose we have two points: A(1, 2, 3) and B(4, 6, 8).

• Midpoint M = ((1 + 4)/2, (2 + 6)/2, (3 + 8)/2) = (2.5, 4, 5.5)
• Direction vector D = (4 - 1, 6 - 2, 8 - 3) = (3, 4, 5)

Thus, the normal vector N = (3, 4, 5). Plugging these values into the plane equation:

3(x - 2.5) + 4(y - 4) + 5(z - 5.5) = 0

When expanded, this gives:

3x + 4y + 5z = 43.5

Final Thoughts

This is the equation of the plane that bisects the line segment connecting points A and B perpendicularly. By following these steps, you can find the bisecting plane for any two points in three-dimensional space.