Problem 3 - Calculus: Early Transcendentals

Question

Describe geometrically the set of points \((x, y, z)\) that satisfy \(y=-3 .\)

Step-by-Step Solution

Answer

The set of points is a plane parallel to the xz-plane, with y = -3.

Step 1: Understand the Equation

We have the equation \(y = -3\). This means for every point \((x, y, z)\), the y-coordinate is always \(-3\).

Step 2: Visualize in Three Dimensions

In three-dimensional space, fixing a coordinate (in this case, \(y\)) to a constant value means that the points form a surface, specifically a plane, where the \(y\)-value is always \(-3\).

Step 3: Describe the Set of Points

Because the \(x\) and \(z\) coordinates can be any real number, the set of points is a plane parallel to the xz-plane, but shifted -3 units downward along the y-axis.

Key Concepts

Planes in 3D

In three-dimensional geometry, a plane is a flat surface that extends infinitely in two dimensions. To understand a plane in 3D space, think of it as a sheet of paper without edges that stretches infinitely.
In our context where the equation is \(y = -3\), every point on this plane has a y-coordinate of \(-3\). This means the position across the other two axes, \(x\) and \(z\), can vary freely. By fixing \(y\) at a constant value, we essentially create a slice of the 3D space. This slice or plane is parallel to the xz-plane.

• A plane is defined by fixing one of the three coordinates.
• It extends infinitely along the other two axes.
• In this example, the plane is horizontal due to the fixed \(y\) value.

Recognizing these properties helps grasp how planes can orient themselves within the 3D coordinate system.

Coordinate Systems

A 3D coordinate system adds depth to the familiar 2D axis system, incorporating an additional axis. The axes are typically labeled \(x, y,\) and \(z\). Each coordinate axis is perpendicular to the other two, creating three planes where they intersect. This allows for every point in space to be uniquely identified with a set of three coordinates: \((x, y, z)\).
In the given exercise, the fixed equation \(y = -3\) represents how coordinates can define specific locations or sets in space. When working with coordinate systems:

• The origin \((0, 0, 0)\) is where all three axes intersect.
• Points are expressed as \((x, y, z)\) where each component measures the distance from the origin along the respective axis.
• Planes are formed by fixing one coordinate to a constant value, as seen in the provided example.

Understanding coordinate systems is crucial for navigating and describing 3D space effectively.

Visualizing 3D Shapes

Visualizing shapes in three-dimensional space can initially seem daunting, as it requires comprehending an additional dimension compared to our everyday experience of flat surfaces.
However, with a bit of imagination and practice, picturing concepts like planes becomes second nature. Drawing from our example, once you understand that the equation \(y = -3\) creates a flat plane, you can start to imagine how this shapes the space around you.

• Always start by identifying fixed coordinates which define the plane or surface.
• Visual aids like graphs or 3D models can significantly help.
• Breaking down complex shapes into simpler planes or lines can make them easier to understand.

With tools and techniques to aid visualization, navigating the complexities of 3D shapes becomes a manageable task, enhancing spatial understanding and geometric problem-solving skills.