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Chapter 12: Problem 43

Use intercepts to help sketch the plane. $$ 6 x-3 y+4 z=6 $$

Short Answer

Expert verified
The plane intersects the x-axis at (1, 0, 0), y-axis at (0, -2, 0), and z-axis at (0, 0, 1.5).

Step by step solution

01

Understand the Equation

The equation given is a linear equation in three variables (x, y, and z) that represents a plane in three-dimensional space: \(6x - 3y + 4z = 6\). To sketch this plane, we need to find where it intersects the x, y, and z-axes.
02

Find the X-intercept

To find the x-intercept, set \(y = 0\) and \(z = 0\) in the equation. Substitute these values back into the equation: \(6x - 3(0) + 4(0) = 6\).This simplifies to \(6x = 6\), so \(x = 1\). The x-intercept is (1, 0, 0).
03

Find the Y-intercept

To find the y-intercept, set \(x = 0\) and \(z = 0\) in the equation. Substitute these into the equation: \(6(0) - 3y + 4(0) = 6\).This simplifies to \(-3y = 6\), so \(y = -2\). The y-intercept is (0, -2, 0).
04

Find the Z-intercept

To find the z-intercept, set \(x = 0\) and \(y = 0\) in the equation. Substitute these into the equation: \(6(0) - 3(0) + 4z = 6\).This simplifies to \(4z = 6\), so \(z = 1.5\). The z-intercept is (0, 0, 1.5).
05

Sketch the Plane

Using the intercepts (1, 0, 0), (0, -2, 0), and (0, 0, 1.5), plot these points in a 3D coordinate system and draw lines to connect them. These lines form the triangle that represents where the plane intersects the axes. As a plane is an infinite plane surface, not just the triangle, extend the lines beyond these points knowing the plane continues infinitely.

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Key Concepts

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

Intercepts
Intercepts in mathematics are points where a graph crosses the axes in a coordinate system. In the context of three-dimensional space, a plane may intersect with each of the x, y, and z axes.
These specific intersections are known as the x-intercept, y-intercept, and z-intercept respectively. Finding these intercepts can greatly simplify the task of sketching a plane.
  • **X-intercept**: Found by setting both the y and z variables to zero in the equation, solving for x.
  • **Y-intercept**: Determined by setting x and z to zero, then solving the equation for y.
  • **Z-intercept**: Acquired by setting x and y to zero, leaving z to solve in the equation.
By understanding and finding these intercepts, you can easily visualize where the plane touches each axis in three-dimensional space.
Linear Equation in Three Variables
A linear equation in three variables involves x, y, and z, and typically reflects a plane in three-dimensional space. The general form of this equation is: \[ ax + by + cz = d \]
Where a, b, c, and d are constants.

To solve or graph this equation, you must understand how each variable component interacts. Each pair of variables creates a line, and the entire equation represents a plane.
  • The coefficients (a, b, c) dictate the orientation of the plane.
  • The constant (d) affects the position of the plane relative to the origin.
When you set two of the variables to zero, you highlight where the plane crosses the third axis. These cross-points help visualize the plane's orientation and position in 3D space.
3D Coordinate System
The 3D coordinate system extends the traditional x and y axes found in two-dimensional grids by adding a z axis. Each point in this system is represented by three numbers (x, y, z), describing a location in three-dimensional space.

Visualizing planes and other geometric objects within this framework allows for a deeper understanding and exploration of spatial relationships.
  • **X-axis**: Horizontal, left to right direction.
  • **Y-axis**: Typically vertical, representing up and down movement.
  • **Z-axis**: Adds depth, demonstrating in and out movements relative to the viewer.
These axes intersect at the origin, which is the point (0, 0, 0). Understanding this system is crucial for graphing linear equations in three variables, as it provides the space needed to showcase how planes interact and intersect with each axis.

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Most popular questions from this chapter

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place). $$ 2 x-3 y=z, \quad 4 x=3+6 y+2 z $$

Graph the surfaces \(z=x^{2}+y^{2}\) and \(z=1-y^{2}\) on a common screen using the domain \(|x| \leqslant 1.2,|y| \leqslant 1.2\) and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the \(x y\) -plane is an ellipse.

Find an equation of the plane. The plane through the point \((5,3,5)\) and with normal vector \(2 \mathbf{i}+\mathbf{j}-\mathbf{k}\)

Find an equation of the plane. The plane that passes through the points \((0,-2,5)\) and \((-1,3,1)\) and is perpendicular to the plane \(2 z=5 x+4 y\)

A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet (see the photo on page 839 ). The diameter at the base is \(280 \mathrm{m}\) and the minimum diameter, \(500 \mathrm{m}\) above the base, is \(200 \mathrm{m}\). Find an equation for the tower.
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