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Chapter 9: Problem 71

In Exercises 71 and \(72,\) use a computer algebra system to graph the plane. $$ 2 x+y-z=6 $$

Short Answer

Expert verified
The plane equation 2x + y - z = 6 can be graphed by first converting it into the standard form, \( z = 2x + y - 6 \), and then using a computer algebra system to plot the 3D graph.

Step by step solution

01

Conversion to standard form

The given equation for the plane is 2x + y - z = 6. The typical standard form that can be plotted is z = f(x, y). By rearranging the equation into this form we get the equation as \( z = 2x + y - 6 \).
02

Plotting in 3D

After the equation is in the standard form it can now be inputted into a computer algebra system, that supports 3D plotting. The range for x and y coordinates can be arbitrarily chosen although for best results they often coincide with the values that make \( z = 0 \) in the standard form equation.
03

Analyzing the plot

Lastly, analyze the drawn plot to understand the orientation and position of the plane in the 3D space. The slope of the plane can be interpreted from the coefficients of the x, y, and the constant term in the standard form of the equation.

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Key Concepts

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

Computer Algebra Systems
Computer Algebra Systems (CAS) are powerful tools that help with calculations and graphing in mathematics. They work like advanced calculators that can handle complex mathematical equations. These systems are useful for students, engineers, and scientists.
  • Solving Equations: CAS can solve equations symbolically, which means they solve them in terms of variables rather than numbers. This is vital when plotting 3D graphs because it helps in rearranging the equations into a suitable form.
  • Graphical Representation: They help in graphing 2D and 3D plots of functions and equations. This graphical representation is crucial for understanding how certain equations translate into geometric shapes, like planes in 3D space.
Using CAS to automate these calculations not only saves time but also reduces human error. For the plane equation given, a CAS can take it and effortlessly graph it in 3D after converting the equation into standard form. This enhances learning by providing immediate visual feedback.
3D Plotting
3D plotting is the process of graphing functions or equations that have three variables typically x, y, and z. It helps to visualize complex relationships between these variables and understand how the function behaves in space.
  • Axes Representation: In 3D graphs, the x, y, and z-axes are used to create a coordinate system. Each point in a 3D space is represented by three values, one for each axis.
  • Understanding Depth: A key benefit of 3D plotting is that it shows the depth of different points relative to each other, showcasing how they form a plane, surface, or shape in space.
  • Application in Planes: When plotting the plane with the equation \(z = 2x + y - 6\), the coefficients provide insight into the plane's orientation in 3D space. The ability to visualize the slopes governed by these coefficients helps in comprehending the geometry involved.
Using tools that support 3D plotting, like many Computer Algebra Systems, plotting an equation becomes an intuitive task, helping clarify concepts that might be difficult to grasp in a purely algebraic form.
Standard Form of Planes
The standard form of an equation is essential for graphing planes. For a plane, the standard form typically rearranges the equation so that one variable is isolated on one side, such as \(z = f(x, y)\).
  • Conversion to Standard Form: For the equation \(2x + y - z = 6\), it's standardized by solving for \(z\). This gives us \(z = 2x + y - 6\), making it easy to graph in a 3D environment.
  • Influence of Coefficients: The coefficients of \(x\) and \(y\) indicate the plane's inclination or slope in 3D space. The fact that \(z\) depends directly on \(x\) and \(y\) shows how changes in these variables affect the plane's position.
  • Benefits in Plotting: By converting to standard form, the equation seamlessly integrates into 3D graphing tools. This allows one to directly visualize and interpret the spatial relationships and slopes within the plotted plane.
Understanding the standard form is crucial because it simplifies the plotting process and provides a straightforward path to visualizing algebraic equations in a geometric context.

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Most popular questions from this chapter

In Exercises 61 and \(62,\) sketch the solid that has the given description in cylindrical coordinates. $$ 0 \leq \theta \leq 2 \pi, 2 \leq r \leq 4, z^{2} \leq-r^{2}+6 r-8 $$

In Exercises 45 and \(46,\) the initial and terminal points of a vector \(v\) are given. (a) Sketch the directed line segment, (b) find the component form of the vector, and (c) sketch the vector with its initial point at the origin. Initial point: (-1,2,3) Terminal point: (3,3,4)

Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \\ \mathbf{v}=\mathbf{i}-2 \mathbf{j}+\mathbf{k} \end{array} $$

Think About It In Exercises \(65-68\), find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=9,\) and inside the cone \(z^{2}=x^{2}+y^{2}\)

(a) find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and (b) find the vector component of u orthogonal to v. $$ \mathbf{u}=\langle 2,-3\rangle, \quad \mathbf{v}=\langle 3,2\rangle $$
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