Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 5 x+8 y-2 z=10 $$

Step-by-Step Solution

Answer

The equation represents a plane with intercepts at (2,0,0), (0,5/4,0), and (0,0,-5).

Step 1: Identify the Plane Equation Format

The equation given is \(5x + 8y - 2z = 10\), which is in the standard form of a plane in three-dimensional space, \(Ax + By + Cz = D\). Here, \(A = 5\), \(B = 8\), \(C = -2\), and \(D = 10\).

Step 2: Determine the Intercepts

To sketch the plane, find the x-, y-, and z-intercepts by setting the other two variables to zero. - For the x-intercept, set \(y = 0\) and \(z = 0\): \(5x = 10\) gives \(x = 2\). Thus, the x-intercept is \((2, 0, 0)\).- For the y-intercept, set \(x = 0\) and \(z = 0\): \(8y = 10\) gives \(y = \frac{5}{4}\). Thus, the y-intercept is \((0, \frac{5}{4}, 0)\).- For the z-intercept, set \(x = 0\) and \(y = 0\): \(-2z = 10\) gives \(z = -5\). Thus, the z-intercept is \((0, 0, -5)\).

Step 3: Sketch the Graph

Plot the intercepts calculated in Step 2 on a three-dimensional coordinate system.- Mark the point \((2, 0, 0)\) on the x-axis.- Mark the point \((0, \frac{5}{4}, 0)\) on the y-axis.- Mark the point \((0, 0, -5)\) on the z-axis.Draw lines through the points to form a triangular section of the plane and sketch the plane extending outward in all directions.

Key Concepts

Three-Dimensional Space

Three-dimensional space is an extension of the concept of flat or two-dimensional space. It gives us depth and allows us to model the physical world more accurately. In this space, we work with three axes: the x-axis (horizontal), y-axis (vertical), and z-axis (depth). Each point in this three-dimensional coordinate system is represented by a set of three numbers, \(x, y, z\). These numbers indicate the point's position along each of the three axes.

• The x-axis runs left-right
• The y-axis runs up-down
• The z-axis comes forward-backward

In practical situations, this setup allows us to describe positions and shapes more effectively. For instance, when dealing with equations of planes, as in the exercise above, it becomes crucial to visualize these equations within this three-dimensional context.

Intercepts

Intercepts are the points where a graph intersects the axes. In three-dimensional space, we have three types of intercepts: x-intercept, y-intercept, and z-intercept. To find these intercepts, it is common practice to set the other two variables to zero and solve the equation to find the value of the intercept.

• X-intercept: Found by setting y = 0 and z = 0
• Y-intercept: Found by setting x = 0 and z = 0
• Z-intercept: Found by setting x = 0 and y = 0

By finding these intercepts, you gain crucial information to help graph the plane. As demonstrated in the solution, the intercepts \( (2, 0, 0), \(0, \frac{5}{4}, 0\), \(0, 0, -5\) \) help you visualize where the plane touches each axis. These points are pivotal in sketching the plane.

Graphing in Three-Space

Graphing in three-dimensional space can seem tricky because it's more complex than two-dimensional graphing. However, by breaking it down into parts, like plotting intercepts, it can become more manageable. In three-space, you can still use the Cartesian coordinate system, but you have an extra dimension to consider.
The process involves:

• Determining where the shape intersects with each axis (using intercepts)
• Positioning these points accurately along their respective axes
• Drawing lines or planes through these points

For example, once you have the intercepts on a three-dimensional plane, you draw lines between these points. Connecting them gives you a "slice" of the plane and indicates its direction and slope in space. Always remember that the plane continues infinitely in all directions beyond the points you have plotted.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to represent and analyze geometrical figures. It allows for a more precise understanding and solving of geometric problems through algebraic equations. When dealing with planes, as in the exercise, you can use a general equation of the form \(Ax + By + Cz = D\). Here:

• \(A, B, C\) determine the orientation of the plane
• \(D\) is a constant that shifts the plane in three-dimensional space

Coordinate geometry helps bridge algebra with geometry by providing tools to calculate distances, angles, and positions within space. This practice is essential when trying to understand plane equations and intercepts because it helps visualize how everything comes together in a three-dimensional coordinate system. The clarity it offers can greatly enhance comprehension and problem-solving skills in three-space scenarios.