Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 3 x+2 z=10 $$

Step-by-Step Solution

Answer

The equation \( 3x + 2z = 10 \) represents a plane parallel to the y-axis in three-space, intersecting the xz-plane along the line \( 3x + 2z = 10 \).

Step 1: Identify Variables and Constants

The equation given is \( 3x + 2z = 10 \). We notice that this equation involves only the variables \( x \) and \( z \), and it does not contain the variable \( y \). In three-space, when an equation lacks one variable, it typically represents a plane that extends infinitely in the direction of that missing variable.

Step 2: Determine the Trace in the xz-plane

Since the equation is only dependent on \( x \) and \( z \), it lies within the xz-plane when we set \( y = 0 \). Setting \( y = 0 \) does not change the equation. Therefore, in the xz-plane, the equation \( 3x + 2z = 10 \) describes a line.

Step 3: Establish the Intersection with Coordinate Axes

To sketch the graph, find where it intersects the axes by setting one variable to zero: - Set \( x = 0 \) to find the z-intercept: \( 2z = 10 \Rightarrow z = 5 \). - Set \( z = 0 \) to find the x-intercept: \( 3x = 10 \Rightarrow x = \frac{10}{3} \approx 3.33 \). These intercepts are points on the line in the xz-plane.

Step 4: Sketch the Graph in Three-Space

Since the equation lacks the variable \( y \), the line \( 3x + 2z = 10 \) can be drawn in the xz-plane. This line becomes a plane by extending it parallel along the \( y \)-axis. In three-space, the line forms a plane that runs infinitely parallel to the y-axis, intersecting it at every y-value.

Step 5: Name the Plane

The plane is sometimes referred to by its equation or by the axes it is primarily associated with. In this case, it's a plane parallel to the y-axis, defined mainly by its equation \( 3x + 2z = 10 \). It can also be called an xz-plane parallel plane.

Key Concepts

Three-Space

Three-space is simply a term used to refer to our familiar three-dimensional space. It is where we visualize points and shapes using three coordinates: \( x \), \( y \), and \( z \). Just like how a flat piece of paper helps us plot points using \( x \)- and \( y \)-coordinates in two dimensions, three-space adds another dimension with the \( z \)-coordinate.

This additional dimension gives more depth to our visualizations, enabling us to represent more complex structures like planes, cubes, and spheres.

• The \( x \)-axis usually runs left to right.
• The \( y \)-axis extends up and down.
• The \( z \)-axis moves front to back.

When working with graphs in three-space, we have more freedom and complexity. For example, an equation could represent a plane, a line, or even a more complex surface. This freedom helps us model our world more accurately because in real life, most of our objects exist in three-space.

Equation of a Plane

An equation of a plane in three-space is a linear equation that typically involves three variables, \( x \), \( y \), and \( z \). However, when one variable is missing, the equation still represents a plane, but this plane extends infinitely in the direction of the missing variable.

In our exercise, the equation \( 3x + 2z = 10 \) lacks the \( y \)-variable. This means the plane extends parallel to the \( y \)-axis, covering all possible \( y \)-values.

• The given equation, \( 3x + 2z = 10 \), describes a plane, but can also be viewed as a line within the \( xz \)-plane.
• Since it lacks the \( y \) component, the plane appears as a line when \( y = 0 \), but in reality, it forms a plane in three-space.

When sketching this plane, remember it will look flat in the \( xz \)-plane, but it actually extends straight up and down along the \( y \)-axis. This concept is crucial because recognizing that a two-variable equation extends into three dimensions helps us understand how to plot it correctly in three-space.

Intersections with Coordinate Axes

Understanding intersections with coordinate axes is essential when sketching graphs in three-space. These intersections help us establish reference points, making it possible to visualize the position and orientation of the graph.

For an equation like \( 3x + 2z = 10 \), we can find these intersection points by setting each variable to zero in turn:

• To find the \( x \)-intercept, set \( z = 0 \), leading to \( 3x = 10 \) or \( x = \frac{10}{3} \approx 3.33 \).
• To find the \( z \)-intercept, set \( x = 0 \), leading to \( 2z = 10 \) or \( z = 5 \).

These points provide us with specific coordinates (\( x = 3.33, z = 0 \) and \( x = 0, z = 5 \)) on the \( xz \)-plane. From these intercepts, we can accurately draw a line in the \( xz \)-plane. Then, imagining this line stretches parallel along the \( y \)-axis, we can see it forming a plane in three-space.

By understanding these intersections, we gain a clearer picture of how equations map out in three-space, and appreciate how each variable affects the graph's shape.