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In geometry, understanding the equations of planes is fundamental for analyzing the relations among planes in a three-dimensional space. A plane is a flat, two-dimensional surface that extends infinitely in all directions. The equation of a plane can be expressed in the form of Ax + By + Cz + D = 0, where A, B, and C are the coefficients representing the normal vector, and D is a constant that helps to position the plane in the space relative to the origin.

In the context of parallel planes, it's crucial to note that their normal vectors are identical or scalar multiples of each other. Thus, if we have two parallel planes, the equations might look like Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0, differing only in the constant term. This means they share the same normal vector, given by n = (A, B, C), which is perpendicular to both planes. The different constants, D1 and D2, represent the distances of the planes from the origin along the direction of the normal vector.

Vector projection is another important concept in understanding the relationships between points and planes. Imagine you have a point P lying on one plane, and you want to find its closest point Q on a parallel plane. The vector drawn from P to Q (PQ) is essentially the projection of point P onto the other plane.

Importantly, since the planes are parallel, this vector PQ is parallel to the normal vector of the planes, n. If we know the coordinates of P (through its position vector p) and the equation of the plane on which Q lies, we can effectively use vector projection to find Q's position (vector q). The concept of vector projection not only helps in locating points but is also vital in calculating distances, which brings us to the notion of point-normal form distance.

The point-normal form is an indispensable tool for finding the distance between a point and a plane in three-dimensional space, and it can similarly be applied to find the distance between two parallel planes. This distance is calculated by projecting the vector between any point on one plane, and its closest point on the other plane onto the normal vector of the planes.

When we say that two planes are everywhere equidistant, we mean that the length of the vector PQ (where P is a point on one plane and Q is the projection of P on the other plane) is the same regardless of the location of P. The calculation for this distance involves finding the scalar multiple k such that PQ = kn. Once k is determined, the distance d can be simplified to k|n|, indicating a constant distance between the parallel planes. This concept is critical when proving that parallel planes are indeed equidistant, which is a frequent exercise in geometric studies of space.