91Ó°ÊÓ

When discussing geometric shapes in mathematics, a plane is a two-dimensional flat surface that extends infinitely in three-dimensional space. The mathematical representation of a plane often involves a linear equation with three variables: \(x\), \(y\), and \(z\). In the given exercise, the plane is represented by the equation \(3x - 2y + z = 2\). Here, each coefficient and variable plays a crucial role:

• \(3x\) indicates the influence of the \(x\)-coordinate on the plane.
• \(-2y\) represents how the \(y\)-coordinate affects the plane.
• \(z\) is the direct influence of the \(z\)-coordinate.
• The constant \(2\) determines the plane's position in space relative to the origin.

The balance of these terms creates an infinite set of points satisfying the equation, and collectively, these points form the plane.

Understanding three-dimensional space is fundamental in grasping how planes exist and interact within it. In three-dimensional space, every point is defined by three coordinates \((x, y, z)\). These coordinates describe the point's position relative to the three perpendicular axes, usually referred to as the x-axis, y-axis, and z-axis.In this context:

• X-axis: Runs horizontally and extends left to right.
• Y-axis: Also runs horizontally but extends front to back.
• Z-axis: Runs vertically and extends top to bottom.

A plane, like in the equation \(3x - 2y + z = 2\), slices through this space. It's crucial to visualize this as a flat surface cutting through the volume defined by three-dimensional space, affecting how points on the plane are arranged.

Parametric equations provide a powerful way to describe geometric objects like curves and surfaces. Instead of relying on a single equation involving all variables, parametric equations break down these objects into components, often tied to one or more parameters typically noted as \(s\) and \(t\).For the plane equation \(3x - 2y + z = 2\), we used parameterization to express solutions as a set of separate equations:

• \(x(s, t) = s\) - meaning \(x\) is directly influenced by parameter \(s\).
• \(y(s, t) = t\) - meaning \(y\) corresponds with parameter \(t\).
• \(z(s, t) = 2 - 3s + 2t\) - expressing \(z\) as a function of both parameters.

These parametric descriptions enable a clear and structured way to calculate specific points on the plane. By varying the values of \(s\) and \(t\), you can explore the entire plane's surface, essentially mapping out every possible point that satisfies the original plane equation.