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Chapter 11: Problem 8

Describe and sketch the surface. $$ x=4 $$

Short Answer

Expert verified
The equation \(x = 4\) represents a vertical plane standing on the x-axis at the point (4,0,0) and extending indefinitely in both the positive and negative y and z directions.

Step by step solution

01

Understanding the problem and identifying the type of surface

The equation \(x = 4\) is an equation of a fixed x that does not relate to either y or z. This represents a vertical plane parallel to the yz-plane and the distance to the yz-plane is 4 units. Moreover, this plane includes all possible combinations of y and z values, while x is kept constant.
02

Describing the surface

The surface described by the equation \(x = 4\) is a plane standing vertically. It crosses the x-axis at the point (4, 0, 0), and it is parallel to yz-plane. The plane includes all the points where x is equal to 4, and so it extends infinitely in the y and z directions.
03

Sketching the surface

When sketching, mark these three points: (4, 0, 0), (4, 1, 0), and (4, 0, 1). Connect these 3 points and extend the lines indefinitely. The plane should appear as a vertical line in 2D, but remember that it actually extends infinitely 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.

Equation of a Plane
In mathematics, an equation of a plane is a way to describe a flat surface in three-dimensional space. The general form of a plane equation is \(Ax + By + Cz + D = 0\), where \(A\), \(B\), \(C\), and \(D\) are constants, and \((x, y, z)\) are the coordinates of any point on the plane. For a specific case like \(x = 4\), the equation simplifies as no y or z terms are present. This means the plane is positioned at a constant x-value and does not depend on the values of y and z. This simplification makes such surfaces easier to identify and understand. They reveal where exactly the plane will cut across or align itself in regards to a particular axis within a space.
yz-plane
The yz-plane is one of the three Cartesian coordinate planes, specifically where \(x = 0\) throughout. The yz-plane contains all points where the x-coordinate is zero, effectively making it a flat plane comprising all the y and z values. When dealing with an equation like \(x = 4\), we are talking about a plane that is parallel to the yz-plane. That means the plane at \(x = 4\) does not intersect the yz-plane; instead, it runs alongside it at a distance of 4 units away along the x-axis. So undeniably, understanding the yz-plane helps comprehend how certain planes situate themselves in relation to it.
Vertical Plane
A vertical plane is one which extends up and down infinitely, and generally stays parallel to one of the coordinate planes such as the yz-plane. In the described scenario, the equation \(x = 4\) represents a vertical plane. Such a plane stands upright like a wall鈥攁 visual easier to imagine in physical terms. It is characterized by a fixed x-value and unrestricted, infinite extensions in both y and z directions. Visualizing vertical planes can be beneficial in multiple disciplines, among them architecture, engineering, and the sciences, where visualizing space and structures in three dimensions is crucial.
Coordinate System
Understanding the coordinate system is fundamental when exploring 3D surfaces. A typical 3D Cartesian coordinate system consists of three perpendicular axes: the x-axis, y-axis, and z-axis. Where these axes intersect is called the origin, marked as \((0, 0, 0)\). Each point in this system is defined by an ordered triplet \((x, y, z)\), indicating its position in three-dimensional space. This setup helps in visualizing and plotting complex 3D structures. For instance, a plane with the equation \(x = 4\) will be plotted by fixing x and allowing y and z to vary, creating a plane perpendicular to the x-axis.

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Most popular questions from this chapter

Find the distance between the point and the plane.\((0,0,0)\) \(2 x+3 y+z=12\)

Give a geometric description of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).

Find a unit vector (a) in the direction of \(\mathrm{u}\) and (b) in the direction opposite of \(\mathbf{u}\). \(\mathbf{u}=\langle 8,0,0\rangle\)

Prove the triangle inequality \(\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|\).

The initial and terminal points of the vector \(\mathbf{v}\) are \(\left(x_{1}, y_{1}, z_{1}\right)\) and \((x, y, z)\). Describe the set of all points \((x, y, z)\) such that \(\|\mathbf{v}\|=4\)
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