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Chapter 10: Problem 35

Sketch the given plane. $$x=4$$

Short Answer

Expert verified
An equation of the form \(x = a\) represents a 3D plane parallel to the yz-plane. For the equation \(x=4\), this plane intercepts the x-axis at the point \(x=4\), and extends indefinitely in the y and z directions.

Step by step solution

01

Understand the form of the equation

An equation of the form \(x = a\) represents a plane parallel to the yz-plane in a three dimensional space. Here, \(x = 4\) signifies that the plane intersects the x-axis at \(x=4\).
02

Draw the plane

On a 3D coordinate system, draw a plane parallel to the yz-plane (in other words, the plane should extend in the y and z directions) which intersects the x-axis at \(x=4\). This can be done by first drawing a straight line parallel to the yz-plane at \(x=4\), and then extending this line in both y and z directions to create a plane.
03

Label the plane

To finalize the sketch, label the plane as \(x=4\) and draw arrows along the plane edges to show that it extends indefinitely in the y and z directions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plane Equation
In the realm of the 3D coordinate system, understanding plane equations is essential. A plane in three-dimensional space can be defined using different forms of equations. One common type is the plane equation represented by a constant, such as \(x = a\). This equation means that all points on the plane have an x-coordinate that is always equal to \(a\).\(x = 4\) is an example of such an equation. It defines a set of points making up a flat surface in 3D space, where every point has the x-coordinate fixed at 4. The equation suggests that the plane extends infinitely along the y and z directions, maintaining the x-coordinate constant. Remember that the equation doesn鈥檛 provide any boundary along the y or z axes, allowing the plane to continue indefinitely.
yz-plane
The yz-plane is an important reference in understanding the position and orientation of planes in a 3D coordinate system. The yz-plane itself is where the x-coordinate is zero, described by the equation \(x=0\). This plane extends in the positive and negative directions along the y and z axes.
The equation \(x = 4\) describes a plane parallel to the yz-plane but displaced by 4 units along the x-axis. Hence, the given plane doesn鈥檛 intersect the yz-plane; instead, it floats parallel to it. This means that every point on the plane at \(x = 4\) has an x-coordinate of 4, with y and z coordinates being any real number.
  • The plane extends without limits along the y and z directions.
  • It is like sliding the yz-plane 4 units away from the origin along the x-axis.
x-axis intersection
The intersection of a plane with the x-axis is crucial in visualizing its orientation in 3D space. When a plane is described by an equation like \(x = 4\), its intersection with the x-axis happens precisely at this x-coordinate.
In this specific case, the plane intersects the x-axis at the point (4, 0, 0). This intersection point is the only place where the plane touches the x-axis, as the plane extends in the y and z directions indefinitely.
This intersection confirms the plane's orientation parallel to the yz-plane and its position in 3D space, providing a foundational understanding needed to sketch or visualize the plane effectively.
  • The plane \(x=4\) never passes through any point on the x-axis other than (4, 0, 0).
  • This characteristic helps in understanding and sketching the plane's position easily.

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