91Ó°ÊÓ

The midpoint formula is a fundamental tool in coordinate geometry used to find the center point between two given points in a three-dimensional space. Simply, it is the point that is exactly halfway between the two points. To calculate the midpoint, you can use the formula:
\[\left( x_m, y_m, z_m \right) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\]
where

• \((x_1, y_1, z_1)\) are the coordinates of the first point,
• \((x_2, y_2, z_2)\) are the coordinates of the second point,
• \((x_m, y_m, z_m)\) represents the midpoint coordinates.

For example, given points \((4, -2, 3)\) and \((2, 4, -1)\), the midpoint is calculated as:
\( \left( \frac{4 + 2}{2}, \frac{-2 + 4}{2}, \frac{3 + (-1)}{2} \right) = (3, 1, 1) \).
This midpoint, \((3, 1, 1)\), is used in further calculations as it lies on the bisecting plane.

Direction vectors are key in understanding the orientation of a line in space. They are derived from the difference in coordinates between two points on the line. A direction vector helps to determine how one point leads to another in a straight path.
For a line joining points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the direction vector \(\vec{d}\) is:
\[ \vec{d} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \] In the given exercise, the direction vector from points \((4, -2, 3)\) to \((2, 4, -1)\) is calculated as:
\[ \vec{d} = (2 - 4, 4 - (-2), -1 - 3) = (-2, 6, -4) \].
This vector essentially describes the direction and rate of change as one moves from the first point to the second point. It is also integral in determining the normal vector for the plane.

The equation of a plane in three-dimensional space can be formulated when you know a point on the plane and a vector that is perpendicular (or normal) to the plane. This provides a way to describe any plane mathematically.
To construct this equation, you need:

• A normal vector \((a, b, c)\), which is perpendicular to the plane.
• A point \((x_0, y_0, z_0)\) through which the plane passes.

The general form of the plane equation is:
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]
Using the direction vector \((-2, 6, -4)\) as the normal vector, and the midpoint \((3, 1, 1)\) as a point on the plane, we derive:
\[-2(x - 3) + 6(y - 1) - 4(z - 1) = 0\].
Simplifying this equation gives:
\[2x - 6y + 4z = 4\].
This is the equation of the plane that bisects the line joining the two given points at right angles and includes the midpoint of the line as well.