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Chapter 11: Problem 15

Sketch the following planes in the window \([0,5] \times[0,5] \times[0,5]\) $$x=2$$

Short Answer

Expert verified
Question: Sketch the plane x = 2 in the window [0,5] 脳 [0,5] 脳 [0,5] and briefly describe the resulting image. Answer: The resulting image of the plane x = 2 in the given window is a vertical rectangle positioned at x=2, spanning the full range of the y and z axes, within the 3D coordinate system. This rectangular plane intersects the coordinate axes at the points (2, 0, 0), (2, 5, 0), and (2, 0, 5).

Step by step solution

01

Setting up the coordinate axes in the given range.

To begin, let's draw three lines to represent the x, y, and z axes. Ensure that each line begins at the origin (0, 0, 0) and spans the given range, which means that they should end at the points (5, 0, 0), (0, 5, 0), and (0, 0, 5), respectively.
02

Locating the intersection points to the coordinate axes

For the plane x = 2, the intersections to the coordinate axes are (2, 0, 0), (2, 5, 0), and (2, 0, 5). These are the points where the plane intersects the x, y, and z axes in the given window. With these points, we can now draw lines connecting them to form the plane.
03

Drawing the boundary lines connecting the intersection points

Connect the intersection points to form a rectangle. A line between (2, 0, 0), and (2, 5, 0) will form the bottom edge of the plane, and another line between (2, 0, 0) and (2, 0, 5) will form the left edge. Connect (2, 5, 0) and (2, 0, 5) to create the diagonal that will help us complete the rectangle. Draw the right and top edges by connecting (2, 5, 0) to (2, 5, 5) and (2, 0, 5) to (2, 5, 5), respectively.
04

Shading the plane

Finally, to complete the sketch, shade the rectangular area formed by these boundary lines to create the plane x = 2 in the window [0,5] 脳 [0,5] 脳 [0,5]. Once you have completed these steps, you will have a sketch of the plane x = 2 in the given window. The plane will be visible as a vertical rectangle in the 3D coordinate system, positioned at x=2 and spanning the full range of the y and z axes.

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Key Concepts

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

Coordinate Axes
In 3D graphing, the coordinate axes are the building blocks upon which we construct our visual representation of three-dimensional space. Picture the axes as a set of intersecting lines that divide space into different sections. There are three axes to consider: the x-axis (usually horizontal from left to right), the y-axis (vertical), and the z-axis (coming out towards you). Each axis is labeled with both positive and negative numbers, with the point where they all intersect marked as the origin (0, 0, 0).

When sketching planes, as with the exercise example plane at x=2, it's essential to start by drawing the coordinate axes within the given range. For instance, if our range is from 0 to 5, each axis would be represented by a line or arrow that starts at the origin and extends up to 5 in their respective positive directions before stopping, forming a kind of 3D box corner.
Intersection Points
The concept of intersection points is key in visualizing how a plane interacts with the coordinate axes in 3D space. These points are the locations where the plane crosses the axes. With the example x=2, the plane intersects the coordinate axes at specific points: (2, 0, 0) on the x-axis, (2, 5, 0) on the y-axis, and (2, 0, 5) on the z-axis. Identifying these points is an essential step since they act as 'anchors' that can be connected to define the edges of the plane.

To improved understand this, consider the plane as a flat sheet of paper that is slicing through our 3D box. Each intersection point marks a corner of this 'paper', and by drawing lines between them, we create a bounded shape that represents the plane.
3D Graphing
The process of 3D graphing involves plotting points, lines, and planes within a three-dimensional coordinate system. It's similar to graphing in two dimensions, but with the added complexity of depth. When creating a graph in 3D, observe that instead of squares, our grid consists of cubes, better termed as 'voxels' in some disciplines. These help us measure and visualize the position of objects in space.

For students looking to improve their understanding of such concepts, engaging with 3D graphing software can be immensely beneficial. These tools allow for the rotation and manipulation of 3D structures, which reinforces the student's spatial reasoning abilities. When graphing our example plane x=2, we visualize it as a flat surface that does not bend or curve, standing upright parallel to the yz-plane.
Calculus Visualization
Calculus is a branch of mathematics that frequently requires the representation of functions and equations in a visual form. Calculus visualization is about taking the abstract concepts and seeing them in a way that makes them easier to comprehend and analyze. For instance, when considering derivatives or integrals in three dimensions, sketching curves and surfaces can help to understand the rate of change or the total accumulation within a region of space.

Considering our example plane x=2, in calculus, this plane could represent a constant value for any function that depends solely on the x-coordinate. This visualization can assist with comprehending topics such as gradients, where we'd imagine how this plane would 'touch' or 'cross' a three-dimensional surface and which direction it slopes or rises. Thus, embedding the concept of planes within the full context of 3D calculus can enhance a student's overall grasp of the subject matter.

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Most popular questions from this chapter

Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). a. Assume that \(\lim \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle,\) which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0 .\) Prove that \(\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \quad\) and \(\quad \lim _{t \rightarrow a} h(t)=L_{3}\). b. Assume that \(\lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2},\) and \(\lim _{t \rightarrow a} h(t)=L_{3} .\) Prove that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle\) which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\).

Distance between a point and a line in the plane Use projections to find a general formula for the (least) distance between the point \(\left.P\left(x_{0}, y_{0}\right) \text { and the line } a x+b y=c . \text { (See Exercises } 62-65 .\right)\).

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\frac{t}{t^{2}+1} \mathbf{i}+t e^{-t^{2}} \mathbf{j}-\frac{2 t}{\sqrt{t^{2}+4}} \mathbf{k} ; \mathbf{r}(0)=\mathbf{i}+\frac{3}{2} \mathbf{j}-3 \mathbf{k}$$

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{-t} \mathbf{i}+2 e^{2 t} \mathbf{j}-4 e^{t} \mathbf{k}\right) d t$$

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and \(f\) are real numbers. It can be shown that this curve lies in a plane. Find a general expression for a nonzero vector orthogonal to the plane containing the curve. $$\begin{aligned}\mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\ &+(e \cos t+f \sin t) \mathbf{k},\end{aligned}$$ where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).
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