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The point-normal form is a key method to determine the equation of a plane in three-dimensional space. It involves using a given point on the plane and the plane's normal vector. The normal vector is perpendicular to the plane's surface, guiding the orientation of the plane. To find the point-normal form of a plane, use the equation:\[ (\vec{p} - \vec{p_0}) \cdot \vec{n} = 0 \]Here:

• \( \vec{p} \) represents any point on the plane.
• \( \vec{p_0} \) is a specific given point on the plane.
• \( \vec{n} \) is the normal vector of the plane.

In our example, the point \( (2, 4, 2) \) provides \( \vec{p_0} \), and the normal vector \( (0, 0, 1) \) is aligned with the z-axis, since the plane we are dealing with is parallel to the xy-plane. The point-normal form allows us to determine the plane's equation as \( z = 2 \). It effectively captures the plane's flat nature through its mathematical properties.

Parallel planes maintain equal distance from one another at every point, and their normal vectors are identical or scalar multiples. In three-dimensional space, knowing that two planes are parallel simplifies many calculations. Consider the xy-plane, which is defined by the equation \( z = 0 \). A plane parallel to the xy-plane differs only in the z-value, meaning it retains a normal vector of \( (0, 0, 1) \). This is because only the height changes, ensuring planes remain equidistant at all points. If we know a plane is parallel to the xy-plane and passes through a point like \( (2, 4, 2) \), the equation automatically adjusts to \( z = 2 \). This highlights the simplicity of determining the equations for parallel planes, focusing primarily on the constant change in the z-coordinate.

The xy-plane is a fundamental plane in three-dimensional geometry. It's a flat surface extending infinitely along the x and y axes. Its equation is \( z = 0 \), signifying all points on this plane have a zero z-coordinate.Points on the xy-plane, like \( (x, y, 0) \), reveal that any elevation (or descent) along the z-axis changes the plane, creating a parallel plane above or below. For example, the plane \( z = 2 \) is parallel to the xy-plane but sits two units above it.This fundamental understanding helps visualize 3D space better:

• The xy-plane is the base reference for many spatial problems.
• Changes in z-values affect plane positioning without altering planar orientation.

By understanding the xy-plane equation, you can grasp how planes relate spatially and compute their equations when parallel conditions are known.