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Chapter 11: Problem 65

Label any intercepts and sketch a graph of the plane.\(2 x-y+3 z=4\)

Short Answer

Expert verified
The intercepts are at (2,0,0) for x, (0,-4,0) for y, and (0,0,4/3) for z. These should be marked on the 3D graph and a plane drawn passing through them.

Step by step solution

01

Determine the X-Intercept

Set \(y\) and \(z\) to 0, and solve for \(x\). This will yield the x-intercept.\(2x - 0 + 3*0 = 4\)\r\nSo, \(x = 2\). The x-intercept is (2,0,0).
02

Determine the Y-Intercept

Set \(x\) and \(z\) to 0, and solve for \(y\). This would reveal the y-intercept.\(-2*0 - y + 3*0 = 4\)\r\nSo, \(y = -4\). Therefore, the y-intercept is (0,-4,0).
03

Determine the Z-Intercept

Set \(x\) and \(y\) to 0, and solve for \(z\) to get the z-intercept.\r\n\(2*0 - 0 + 3z = 4\)\r\nSo, \(z = 4/3\). Hence, the z-intercept is (0,0,4/3).
04

Sketch the Plane

Plot the three intercepts on the 3D coordinate system and draw the plane that passes through them.
05

Label the Intercepts

Finally, mark the intercepts on the graph.

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Key Concepts

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

x-intercept Calculation
To find the x-intercept of a plane in a three-dimensional space, you need to determine the point where the plane crosses the x-axis. This is done by setting the y and z coordinates to zero and solving for x. For the equation 2x - y + 3z = 4, you set y and z to zero, resulting in 2x = 4, which simplifies to x = 2.

The x-intercept is then presented as a point in the 3D coordinate system, specifically (2,0,0). At this point, the plane crosses the x-axis, and these coordinates are used to sketch the graph of the plane.
y-intercept Calculation
When you want to calculate the y-intercept for a plane, it involves finding where the plane will intersect the y-axis. Similar to the process of finding the x-intercept, you will solve for y by setting the values of x and z to zero. In our example equation 2x - y + 3z = 4, by assigning x and z as zero, you get -y = 4, which gives y = -4.

This provides us with the y-intercept point (0,-4,0), meaning that the plane intersects the y-axis at the point four units in the negative y-direction.
z-intercept Calculation
The z-intercept is located where the plane intersects the z-axis. To find the z-intercept, both x and y are set to zero, and the equation is then solved for z. Using the given plane equation 2x - y + 3z = 4, after setting x and y to zero, we are left with 3z = 4, and z becomes z = 4/3.

The resulting z-intercept is the point (0,0,4/3). This shows us that the plane meets the z-axis at a height of 4/3 units above the origin.
3D Coordinate System
Understanding the 3D coordinate system is essential for graphing planes in three-dimensional space. The system consists of three axes: the x-axis, y-axis, and z-axis, which are all perpendicular to one another. Every point in this space can be represented by a set of three coordinates (x,y,z).

When sketching a plane, such as 2x - y + 3z = 4, you start by plotting the intercepts calculated previously. The x-intercept (2,0,0), y-intercept (0,-4,0), and z-intercept (0,0,4/3) mark where the plane touches the respective axes. Once these points are plotted, you can draw a flat surface that extends through them, representing the plane in the coordinate system.

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Most popular questions from this chapter

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