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Chapter 12: Problem 3

Describe the plane \(x=4\)

Short Answer

Expert verified
Question: Explain the location and appearance of the plane x=4 in the Cartesian coordinate system. Answer: The plane x=4 is a vertical plane parallel to the yz-plane. It is situated 4 units away from the yz-plane along the positive x-axis. The plane extends infinitely in the directions of the y- and z-axes and intersects the x-axis at the point (4,0,0).

Step by step solution

01

Understand the given equation.

The equation of the plane is given as \(x=4\). This is a very simple equation because it only involves one variable (x-coordinate). Since there are no terms with y or z, we can infer that the position of the plane does not depend on the y- and z-axis.
02

Determine the plane's relation to the Cartesian coordinate system.

As the equation of the plane is given as x=4, it means that every point on the plane has an x-coordinate of 4. So, the plane is parallel to the yz-plane and has a fixed distance from the origin. The plane is 4 units away from the origin along the x-axis.
03

Describe the plane's appearance.

The plane \(x=4\) is a vertical plane parallel to the yz-plane and situated 4 units away from the yz-plane along the positive x-axis. The plane extends infinitely in the directions of the y- and z-axes.
04

Visual representation of the plane.

To visualize the plane, imagine the Cartesian coordinate system with the axes x, y, and z. Next, draw a square grid parallel to the yz-plane situated at x=4. You will notice this plane is parallel to the yz-plane and 4 units away from it along the positive x-axis direction. The plane appears as a vertical sheet that extends infinitely in the +y and +z directions, as well as in the -y and -z directions. The plane intersects the x-axis at the point (4,0,0).

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