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Chapter 11: Problem 52

In Exercises 51-56, plot the intercepts and sketch a graph of the plane. \(2x-y+4z=4\)

Short Answer

Expert verified
The intercepts are (2,0,0), (0,-4,0), and (0,0,1). Once these are plotted on a three-dimensional graph, a plane passing through these intercepts can be sketched as the solution to the equation.

Step by step solution

01

Find the intercepts

To find the x-intercept, set y and z equal to zero and solve for x in the given equation, \[2x-y+4z=4\]. Follow the same process to find the y and z intercepts.
02

Plot the intercepts

Plot these points on a three-dimensional graph. This will provide a basic structure for the plane.
03

Sketch the Plane

Using the intercepts as reference points, sketch a plane which goes through all three points. This plane represents the solution to the given equation.

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Key Concepts

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

Understanding the x-intercept
In 3D coordinates, the **x-intercept** is where the graph of an equation intersects the x-axis. At this point, both the y and z coordinates equal zero. To find the x-intercept for the equation \(2x - y + 4z = 4\), we set \(y = 0\) and \(z = 0\):
  • Substituting into the equation: \(2x - 0 + 4(0) = 4\)
  • This simplifies to \(2x = 4\).

Now, solve for \(x\):
  • Dividing both sides by 2 gives \(x = 2\).

Thus, the x-intercept is the point (2, 0, 0). This point is crucial as it shows where the plane intersects the x-axis. It's the spot on the x-axis that the plane makes contact with, serving as one of the anchor points for sketching the 3D plane.
Understanding the y-intercept
The **y-intercept** in 3D coordinates is the point where the graph intersects the y-axis. Here, both the x and z coordinates are zero. To find the y-intercept in the equation \(2x - y + 4z = 4\), we set \(x = 0\) and \(z = 0\).
  • Substituting into the equation: \(2(0) - y + 4(0) = 4\)
  • This simplifies to \(-y = 4\).

Solving for \(y\):
  • Multiply both sides by -1 to find \(y = -4\).

Therefore, the y-intercept is the point (0, -4, 0). It's one of the three key intercepts needed to establish the plane in 3D space. This point represents the location on the y-axis where the plane intersects, thus providing another vital point needed to draw and understand the orientation of the plane.
Understanding the z-intercept
In a three-dimensional space, the **z-intercept** is where the graph intersects the z-axis. This occurs when the x and y values are zero. To find the z-intercept for the equation \(2x - y + 4z = 4\), set \(x = 0\) and \(y = 0\):
  • Plug these into the equation: \(2(0) - 0 + 4z = 4\)
  • This simplifies to \(4z = 4\).

Now solve for \(z\):
  • Divide both sides by 4 to get \(z = 1\).

Thus, the z-intercept is at the point (0, 0, 1). This point is significant as it shows where the plane intersects the z-axis, serving as the third crucial point for drawing the plane. By finding this intercept, we gain a comprehensive view of the plane's orientation, which is essential when plotting the initial framework of the plane within a 3D coordinate system.

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Most popular questions from this chapter

In Exercises 11-20, use the vectors \(\textbf{u}\) and \(\textbf{v}\) to find each expression. \(\quad \quad \quad \textbf{u} = 3\textbf{i} - \textbf{j} + 4\textbf{k} \quad \quad \quad \quad \quad \quad \textbf{v} = 2\textbf{i} + 2\textbf{j} - \textbf{k}\) \((3\textbf{u}) \times \textbf{v}\)

In Exercises 51-54, find the triple scalar product. \(\textbf{u} = \langle 4, 0, 1 \rangle, \textbf{v} = \langle 0, 5, 0 \rangle, \textbf{w} = \langle 0, 0, 1 \rangle \)

In Exercises 31-36, find the general form of the equation of the plane with the given characteristics. Passes through \((0, 2, 4)\) and \((-1, -2, 0)\) and is perpendicular to the \(yz\)-plane

In Exercises 31-36, find a unit vector orthogonal to \(\textbf{u}\) and \(\textbf{v}\). \(\textbf{u} = \textbf{i}+2\textbf{j}\) \(\textbf{v} = \textbf{i}-3\textbf{k}\)

In Exercises 41-46, find a set of parametric equations of the line. (There are many correct answers.) Passes through \((-1, 4, -3)\) and is parallel to \(\textbf{v}=5\textbf{i}-\textbf{j}\)
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