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In coordinate geometry, calculating the distance from a point to a plane involves understanding the spatial arrangement of coordinates. For a point \(x, y, z\), the distance to one of the coordinate planes—XY, YZ, or XZ—is based on the specific coordinate that remains constant in that plane.

Let's break this down:

• **XY-plane:** This plane is characterized by points where the z-coordinate is zero. Therefore, to find the distance from any point \(x, y, z\) to this plane, we focus on the z-coordinate and take its absolute value. For instance, for the point (3, -4, 2), the distance to the XY-plane is \(|2| = 2\).
• **YZ-plane:** Defined by points with an x-coordinate of zero, the distance is determined by looking at the x-value. For the point (3, -4, 2), this becomes \(|3| = 3\).
• **XZ-plane:** Here the critical value is the y-coordinate, as all points in this plane have y equals zero. For our example, the distance is \(|-4| = 4\).

Understanding this concept helps us visualize how a point is positioned relative to the different planes in a three-dimensional space.

Coordinate geometry, also known as analytic geometry, is a blend of geometry and algebra to describe and analyze real-world problems. It allows us to use algebra to model geometric shapes and appreciate the spatial relationships between them. In 3D space, we deal with three axes: x, y, and z, creating a coordinate system that helps locate points and define planes.

In coordinate geometry:

• The **x-axis** extends horizontally.
• The **y-axis** extends vertically.
• The **z-axis** adds depth, coming out towards the observer (or receding away).

A coordinate plane can be seen as a flat surface that extends infinitely along two axes. Understanding the concept of zero coordinate along one axis in these planes enables identifying how far a point is from any plane simply by examining its other coordinate values. This system is an incredibly powerful tool for solving a myriad of problems involving distances, angles, and intersections in space.

Three-dimensional space is the realm in which we live. It allows us to represent and work with objects that have depth, unlike the flat surfaces of two-dimensional (2D) planes. In mathematics, 3D space is defined by three mutually perpendicular axes: x, y, and z.

The three-dimensional coordinate system is foundational for visualizing geometric figures such as cubes, spheres, and planes. Each point in this space is denoted by a triplet \(x, y, z\), which represents:

• **x:** Horizontal position
• **y:** Vertical position
• **z:** Depth position

The concept of planes—XY, YZ, XZ—is crucial as it allows for an easier understanding of how objects and points position relative to these flat surfaces in space. Comprehensively grasping 3D space and its properties is critical for fields like physics, engineering, and computer graphics, where precise modeling and visualization of objects are required.