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Chapter 11: Problem 41

Describe the graph in three-space of each equation. (a) \(z=2\) (b) \(x=y\) (c) \(x y=0\) (d) \(x y z=0\) (e) \(x^{2}+y^{2}=4\) (f) \(z=\sqrt{9-x^{2}-y^{2}}\)

Short Answer

Expert verified
(a) Horizontal plane, (b) Diagonal plane, (c) Two planes, (d) Three planes, (e) Cylinder, (f) Hemisphere.

Step by step solution

01

Horizontal Plane

The equation \(z=2\) describes a horizontal plane parallel to the \(xy\)-plane, located at a height of \(2\) units along the \(z\)-axis. All points on this plane have their \(z\)-coordinate equal to \(2\).
02

Diagonal Plane

The equation \(x = y\) represents a diagonal plane in three-space, passing through the origin at an angle of 45 degrees to both the \(x\)- and \(y\)-axes. Every point on this plane has equal values for \(x\) and \(y\).
03

Coordinate Planes

The equation \(xy = 0\) describes the union of two coordinate planes: the \(yz\)-plane (where \(x = 0\)) and the \(xz\)-plane (where \(y = 0\)). Points on these planes satisfy the condition that the product \(xy\) is zero.
04

Intersection of Planes

The equation \(xyz = 0\) represents a union of three planes: the \(yz\)-plane (\(x = 0\)), the \(xz\)-plane (\(y = 0\)), and the \(xy\)-plane (\(z = 0\)). Points on any of these planes satisfy one of the variables being zero.
05

Cylindrical Surface

The equation \(x^2 + y^2 = 4\) describes a cylinder with a radius of 2 centered along the \(z\)-axis. The cylinder extends infinitely along the \(z\)-direction with circular cross-sections in the \(xy\)-plane.
06

Hemisphere

The equation \(z = \sqrt{9 - x^2 - y^2}\) forms a hemisphere with a radius of 3 centered at the origin. The hemisphere is above the \(xy\)-plane, forming a dome where \(z\)-values are non-negative.

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Key Concepts

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

Horizontal Plane
In the realm of 3D graphs, a horizontal plane holds a special significance. This concept can be visualized using the equation \(z=2\). Here, the equation conveys a plane that is parallel to the \(xy\)-plane.
What makes it distinct is its positioning at a specific height of 2 units along the \(z\)-axis. In simpler terms, this plane is "floating" in space, above the origin. Consequently, all the points on this horizontal plane share a common trait: their \(z\)-coordinate is always 2.
  • This means that if you know the \(x\) and \(y\) coordinates, you can automatically know the \(z\) coordinate is 2.
  • The plane extends infinitely in the directions of the \(x\) and \(y\) axes.
Such planes are a basic yet vital component in visualizing 3D objects and concepts in calculus.
Diagonal Plane
The diagonal plane represented by the equation \(x = y\) introduces a fascinating feature in three-dimensional space. Unlike horizontal or vertical planes, a diagonal plane slices through the space at an angle.
In this scenario, any given point on this plane has the same value for its \(x\)-coordinate and \(y\)-coordinate. Essentially, it passes through the origin diagonally, making an angle of 45 degrees with both the \(x\)- and \(y\)-axes.
  • Consider points like (1,1) and (2,2); they lie on this plane because their \(x\) and \(y\) coordinates are equal.
  • Unlike a single line, this plane isn't restricted to a single dimension; it stretches infinitely in the space defined by the equality of \(x\) and \(y\).
This concept enhances the understanding of how different planes intersect and project in a 3D environment.
Coordinate Planes
In 3D space, equations like \(xy = 0\) often denote coordinate planes. Specifically here, it signifies the union of two distinct planes.
These are the \(yz\)-plane, where \(x = 0\), and the \(xz\)-plane, where \(y = 0\). This means any point residing on these planes results in the product \(xy\) being zero.
  • On the \(yz\)-plane, since \(x=0\), the points could be like (0,1,2) where only \(y\) and \(z\) have values.
  • Similarly, for the \(xz\)-plane, a point could be (3,0,-1), implying values for \(x\) and \(z\) only.
This reflects how the interaction of different planes in space can be represented through these straightforward equations.
Cylindrical Surface
The notion of a cylindrical surface is an intriguing element in calculus, described in this instance by the equation \(x^2 + y^2 = 4\). This translates to a cylinder centered along the \(z\)-axis.
Think of this cylinder as comprising circles of radius 2 in the \(xy\)-plane that stack vertically throughout the \(z\)-dimension. It continues endlessly along the \(z\)-axis, forming a tubular structure.
  • The center of each circular cross-section sits at the origin, making them symmetric around the \(z\)-axis.
  • Regardless of the \(z\)-coordinate, all horizontal cross-sections are identical circles.
This visualization aids in comprehending surfaces that protrude through spaces without ending, essentially building a foundation in mapping complex surfaces.
Hemisphere in 3D
A hemisphere is a half of a sphere, and it is visually captivating when depicted in 3D space, exemplified by the equation \(z = \sqrt{9 - x^2 - y^2}\). This represents a hemisphere with a radius of 3.
Centered at the origin, it rests above the \(xy\)-plane, creating a dome-like structure.
  • The equation ensures that the \(z\)-values are non-negative, reflecting the upper half of a sphere.
  • When \(x\) and \(y\) are zero, \(z\) reaches its peak value at 3, the radius.
By delving into this concept, you're introduced to a variety of ways surfaces form in space, bolstering a more comprehensive perception of 3D figures in calculus.

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Most popular questions from this chapter

Describe in general terms the following "helical" type motions: (a) \(\mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+t \mathbf{k}\) (b) \(\mathbf{r}(t)=\sin t^{3} \mathbf{i}+\cos t^{3} \mathbf{j}+t^{3} \mathbf{k}\) (c) \(\mathbf{r}(t)=\sin \left(t^{3}+\pi\right) \mathbf{i}+t^{3} \mathbf{j}+\cos \left(t^{3}+\pi\right) \mathbf{k}\) (d) \(\mathbf{r}(t)=t \sin t \mathbf{i}+t \cos t \mathbf{j}+t \mathbf{k}\) (e) \(\mathbf{r}(t)=t^{-2} \sin t \mathbf{i}+t^{-2} \cos t \mathbf{j}+t \mathbf{k}, t>0\) (f) \(\mathbf{r}(t)=t^{2} \sin (\ln t) \mathbf{i}+\ln t \mathbf{j}+t^{2} \cos (\ln t) \mathbf{k}, t>1\)

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