91Ó°ÊÓ

In coordinate geometry, a plane can be defined as a flat, two-dimensional surface that extends infinitely far. To describe a plane mathematically, we use an equation. This equation helps identify all the points that lie on the plane. There are several equation forms for a plane, one of which is quite direct: this involves fixing one of the coordinates to a constant value.
For instance, consider the plane equation\( x = 5 \), which can be denoted as \( \pi_1 \). This equation signifies that for any point to be on this plane, its x-coordinate must be 5, while the y and z coordinates can assume any value. Thus, the plane is parallel to the yz-plane (the plane formed by the y- and z-axes).
Similarly, the equation \( y = 6 \), or \( \pi_2 \), means every point on this plane must have a y-coordinate of 6. Again, x and z can be any value, hence this plane is parallel to the xz-plane. Understanding these concepts helps simplify analyzing where a point could lie in space.

Coordinates provide a way to identify the location of points in a plane or space. In the three-dimensional coordinate system, a point is specified with an ordered triplet \(x, y, z\), representing its distance along the x-, y-, and z-axes respectively.
This coordinate framework allows one to designate positions in a plane or three-dimensional space easily. Imagine a piece of graph paper or a 3D graphing tool showing grid lines; each point can be indicated by how far it stretches along these imaginary rulers.

• For the x-coordinate, think about moving left or right.
• For the y-coordinate, consider moving up or down.
• The z-coordinate involves going inwards or outwards, adding depth to the location.

This system helps delineate spatial awareness, and when combined with plane equations, you can precisely locate where a point might intersect with a plane.

Determining if a point lies on a particular plane involves checking whether the point's coordinates satisfy the equation defining that plane.
To decide if a point \( P(5, -3, -3) \) lies on a given plane, we substitute its coordinates into the plane's equation:

• For plane \( \pi_1: x = 5 \), since the point \( P \) has an x-coordinate of 5, it satisfies this equation. Therefore, \( P \) lies on this plane.
• For plane \( \pi_2: y = 6 \), the point \( P \) has a y-coordinate of -3, which does not meet the requirement \( y = 6 \). Thus, \( P \) cannot be on \( \pi_2 \).

Understanding this principle is crucial when analyzing point locations relative to planes, as it helps determine geometric configurations and relationships in space. It not only aids in solving math problems but also forms the basis of many practical applications in engineering and science where spatial placement is key.