91Ó°ÊÓ

In coordinate geometry, the "normal vector" plays a vital role when dealing with planes. A normal vector

• is perpendicular to a given plane.
• Helps in defining the orientation of the plane.

Understanding the normal vector is key when working with planes. In the context of this exercise, the normal vector is derived from the vector pointing from the origin to point \( A(1, -2, 1) \). Here, the vector \( \mathbf{n} = \langle 1, -2, 1 \rangle \) represents the direction that is perpendicular to the plane. It's like a flagpole sticking straight up from a flag, showing which way the plane is oriented in 3D space. This perpendicular relationship is crucial because it helps to fix the plane in space uniquely.

The "point-normal form" of a plane's equation is a simple yet powerful way to describe planes in three-dimensional space. It leverages the normal vector and a specific point on the plane. The typical form of the point-normal equation is:

• \( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \)

Here, \( (x_0, y_0, z_0) \) is a specific point on the plane, and \( \langle a, b, c \rangle \) is the normal vector. For our specific problem:

• The normal vector \( \langle 1, -2, 1 \rangle \)
• The point \( (1, -2, 1) \)

These values substituted in, yield the point-normal form equation \( 1(x - 1) - 2(y + 2) + 1(z - 1) = 0 \). This form allows for the equation to be both compact and descriptive, making it easy to see how the plane interacts with its surroundings.

"Coordinate Geometry" is the study of geometric figures through the use of a coordinate system, typically involving the cartesian coordinates in three dimensions for problems involving planes. Using this system, we can:

• Define points in space using coordinates \((x, y, z)\)
• Specify lines and vectors using directional numbers.

In the problem at hand, coordinate geometry provides us with a way to represent complex 3D relationships in a straightforward mathematical format. From the given problem, it becomes possible to define the equation of a plane through points and vectors easily, thanks to the consistent and logical framework set by coordinate geometry. This framework allows us to derive the final equation \( x - 2y + z = 6 \), reflecting the relationship between its normal vector and a specific point, seamlessly placing it in 3D space.