Problem 10 - Algebra and Trigonometry

Question

\(7-10\) . Describe and sketch the surface represented by the given equation. \(y=-1\)

Step-by-Step Solution

Answer

The equation \(y = -1\) represents a plane parallel to the \(xz\)-plane, shifted 1 unit down along the \(y\)-axis.

Step 1: Understand the Equation

The given equation is a simple linear equation in three-dimensional space. It represents a plane where the variable \(y\) is constant and equal to \(-1\).

Step 2: Identify the Variables and Axes

In three-dimensional space, we typically have three axes: \(x\), \(y\), and \(z\). With \(y = -1\), the equation does not depend on \(x\) or \(z\), meaning these variables can take any value.

Step 3: Describe the Surface

The surface represented by the equation \(y = -1\) is a plane parallel to the \(xz\)-plane. It is offset by \(-1\) unit along the \(y\)-axis. This plane stretches infinitely in the \(x\) and \(z\) directions.

Step 4: Visualize the Sketch

To sketch this plane, draw the \(x\), \(y\), and \(z\) axes. The plane will intersect the \(y\)-axis at \(-1\). It is a flat surface that extends parallel to the \(xz\)-plane, between the \(x\)-axis and \(z\)-axis lines at any point where \(y = -1\).

Key Concepts

Linear Equations

Linear equations are equations in which each term is either a constant or the product of a constant and a single variable. These equations are simple and straightforward, often used as the building blocks for more complex mathematical models. In the context of 3D coordinate systems, a linear equation, such as the one we have here, typically has the form of

• \[ ax + by + cz = d \]

This represents planes in three dimensions, where \(a\), \(b\), and \(c\) are coefficients that determine the orientation of the plane, and \(d\) represents the position.
In our original exercise, the equation \(y = -1\) is a linear equation that simplifies the concept further. Here, the coefficient of \(x\) and \(z\) is zero, indicating that there is no dependency on those variables within the equation. This results in a plane that is fixed in position along the \(y\)-axis, indicating an infinite line of solutions that plot a surface known as a plane in 3D space.

Planes in 3D

A plane in 3D space is a flat, two-dimensional surface that extends infinitely in two directions. In mathematics, it's essentially the 3D equivalent of a two-dimensional line. Planes are defined in 3D space by linear equations of the form \(Ax + By + Cz = D\).
The special quality of planes is their capacity to slice through the three-dimensional coordinate system, creating a boundary that separates space into different regions. When a plane is defined, particular attention is paid to its normal vector, which is perpendicular to the plane, and its intercept with the coordinate axes.

• In the given problem, the equation \(y = -1\) specifies a plane that is parallel to the \(xz\)-plane.
• This results from the absence of \(x\) and \(z\) in the equation, showing that the plane never tilts towards any other axis.
• The intercept here is where the \(y\)-axis meets the plane at \(-1\).

Understanding how such a plane behaves in 3D helps visualize many geometric and algebraic problems.

Visualizing Surfaces in 3D

Visualizing surfaces in 3D can initially seem tricky, but it's a fascinating part of understanding mathematical surfaces like planes. When we talk about surfaces in a 3D space, we refer to regions of the space that a specific equation defines. For visualization, it helps to think of surfaces as the 'skin' that wraps around or slices through the 3D space.
With the equation \(y = -1\), we can imagine the surface as a sheet of paper stretching infinitely,

• never crossing or rotating along the other axes except for the fixed position at \(-1\) on the \(y\)-axis.
• To get a palpable understanding, envision hovering above this paper (the plane) and recognizing a vast unending field.
• On a sketch, this would appear as a straight horizontal line on the \(xz\) plane—because it extends infinitely along both \(x\) and \(z\).

Skilled visualization of such surfaces is not only crucial in fields like physics and engineering but also plays a role in computer graphics, where 3D rendering is a key task. Hence, embracing these concepts arms students with tools to tackle advanced challenges in a robust way.