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Chapter 12: Problem 18

Sketch the following planes in the window \([0,5] \times[0,5] \times[0,5]\) $$z=y$$

Short Answer

Expert verified
Question: Sketch the plane z=y in the window [0,5] x [0,5] x [0,5] and identify the points of intersection with the axes. Answer: The plane z = y has been sketched by identifying and connecting the intersection points on the window boundaries, (0,0,0) and (0,5,5). The points of intersection with the axes are as follows: - z-axis: (0, 0, z) where z is any value within [0, 5] - y-axis: (0, y, 0) where y is any value within [0, 5] - x-axis: (x, 0, 0) where x is any value within [0, 5] In this case, points (0,0,0) and (0,5,5) are the most relevant intersection points as they help in sketching the given plane.

Step by step solution

01

Identify points of intersection with the axes

(0,0,z) for z-axis, (0,y,0) for y-axis, and (x,0,0) for x-axis are the points of intersection of the plane with the respective axes. For the given plane (z=y), y-axis: z = 0 x-axis: x can be any value within the given range in [0,5] z-axis: z=y
02

Identify intersection points on the window boundaries

The given window is a cube with edges of length 5. To identify the intersection points on the window boundaries, we can plug in the values of x, y, and z within the interval [0,5] in the equation of the plane (z=y). y=0: z=0 y=5: z=5 The two intersection points on the window boundaries are (0,0,0) and (0,5,5).
03

Connect points to sketch the plane

Now that we have the two intersection points, we can connect these points to sketch the plane in a three-dimensional coordinate system. To sketch the plane, do the following: 1. Mark the origin (0,0,0). 2. Draw the x, y, and z axes. 3. Mark the two intersection points on the y and z axes: (0,0,0) and (0,5,5). 4. Connect these points to form a plane in the window [0,5] x [0,5] x [0,5]. The plane z=y has now been sketched in the given window.

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Key Concepts

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

Three-dimensional coordinate system
A three-dimensional coordinate system extends the familiar two-dimensional graph with an additional z-axis. Imagine a room where:
  • The x-axis runs from left to right.
  • The y-axis extends from front to back.
  • The z-axis extends from bottom to top.
Each point in this system is described by three coordinates:
  • x-coordinate: distance along the x-axis.
  • y-coordinate: distance along the y-axis.
  • z-coordinate: height from the ground or base level.
The origin of this system is the point (0,0,0), where all three axes intersect. Understanding this setup is crucial if you want to visualize geometric objects, such as planes, in three dimensions. As you navigate these planes, remember that each point on a plane satisfies its equation simultaneously.
Intersection points
Intersection points are where a plane meets or crosses particular lines, such as the axes or the boundaries of a designated window like a cube. For instance, in the problem of sketching the plane given by the equation z = y, identifying these points helps us outline the plane's position in space.
The intersection with:
  • The y-axis occurs when z equals zero, resulting in any point (0, y, 0) with y values inside the designated interval; indicating that as long as y has value, z has the same value in the given interval [0,5].
  • On the x-axis, theoretically, x could be any number, as the equation z = y doesn't impose restrictions, just establish a relationship between y and z.
  • On the plane z = y itself, we see each point where y and z are equal within the boundaries given, like on the plane itself z=y and y varies from 0 to 5.
Understanding where and how these intersections occur is crucial to accurately represent the plane's boundaries in a given space like [0,5] x [0,5] x [0,5].
Graphing planes
Graphing planes in three dimensions involves more than just drawing lines; it requires us to visualize how these planes extend across the three-dimensional space. With our equation z = y, we are essentially looking at a scenario where every point on the plane matches this criterion.
To graph this plane:
  • Mark the origin, (0,0,0), the central point from which measurements for x, y, and z extend.
  • You should already know how axes are positioned in the three-dimensional system鈥攜et, examine the areas of interaction like between the y and z axis through points like (0,5,5).
  • Connect these points along the x-axis鈥攚hile each y and z coordinates imply the plane stretches infinitely far in both these dimensions whenever constraints like [0,5] are not overriding.
The essence of graphing planes like z = y in a given space is in translating mathematical equations into spatial visualizations, enabling us to deeply experience the laid out path of that equation within the specified window or cube, in this case [0,5] x [0,5] x [0,5].

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Most popular questions from this chapter

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is a circle (in a plane)? b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is an ellipse (in a plane)?

Parabolic trajectory Consider the parabolic trajectory $$ x=\left(V_{0} \cos \alpha\right) t, y=\left(V_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where \(V_{0}\) is the initial speed, \(\alpha\) is the angle of launch, and \(g\) is the acceleration due to gravity. Consider all times \([0, T]\) for which \(y \geq 0\) a. Find and graph the speed, for \(0 \leq t \leq T.\) b. Find and graph the curvature, for \(0 \leq t \leq T.\) c. At what times (if any) do the speed and curvature have maximum and minimum values?

Determine whether the following statements are true and give an explanation or counterexample. a. The line \(\mathbf{r}(t)=\langle 3,-1,4\rangle+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The curve \(\mathbf{r}(t)=\left\langle e^{-t}, \sin t,-\cos t\right\rangle\) approaches a circle as \(t \rightarrow \infty\). d. If \(\mathbf{r}(t)=e^{-t^{2}}\langle 1,1,1\rangle\) then \(\lim _{t \rightarrow \infty} \mathbf{r}(t)=\lim _{t \rightarrow-\infty} \mathbf{r}(t)\).

Maximum curvature Consider the "superparabolas" \(f_{n}(x)=x^{2 n},\) where \(n\) is a positive integer. a. Find the curvature function of \(f_{n},\) for \(n=1,2,\) and 3 b. Plot \(f_{n}\) and their curvature functions, for \(n=1,2,\) and 3 and check for consistency. c. At what points does the maximum curvature occur, for \(n=1,2,3 ?\) d. Let the maximum curvature for \(f_{n}\) occur at \(x=\pm z_{n} .\) Using either analytical methods or a calculator determine \(\lim _{n \rightarrow \infty} z_{n}\) Interpret your result.

Let \(\mathbf{v}=\langle a, b, c\rangle\) and let \(\alpha, \beta\) and \(\gamma\) be the angles between \(\mathbf{v}\) and the positive \(x\) -axis, the positive \(y\) -axis, and the positive \(z\) -axis, respectively (see figure). a. Prove that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) b. Find a vector that makes a \(45^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? c. Find a vector that makes a \(60^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? d. Is there a vector that makes a \(30^{\circ}\) angle with i and \(\mathbf{j}\) ? Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?
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