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Chapter 12: Problem 21

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$3 x-2 y+z=6$$

Short Answer

Expert verified
Question: Find the intersection points of the plane 3x - 2y + z = 6 with the coordinate axes and sketch the graph of the plane. Answer: The intersection points of the plane with the coordinate axes are (2, 0, 0) for the x-axis, (0, -3, 0) for the y-axis, and (0, 0, 6) for the z-axis. When sketching the graph of the plane, connect the intersection points to create a triangular representation within the 3D coordinate system.

Step by step solution

01

Find intersection points with coordinate axes

To find the intersection points with the coordinate axes, we will set two variables to 0 and solve for the remaining variable in the equation of the plane: - To find the intersection with the x-axis (y=0, z=0): $$3x - 2(0) + (0) = 6$$ $$3x = 6$$ $$x = 2$$ So the intersection with the x-axis is \((2, 0, 0)\). - To find the intersection with the y-axis (x=0, z=0): $$3(0) - 2y + (0) = 6$$ $$-2y = 6$$ $$y = -3$$ So the intersection with the y-axis is \((0, -3, 0)\). - To find the intersection with the z-axis (x=0, y=0): $$3(0) - 2(0) + z = 6$$ $$z = 6$$ So the intersection with the z-axis is \((0, 0, 6)\).
02

Find equations of the lines where the planes intersect the coordinate planes

We will find the equations of the lines where the planes intersect the coordinate planes (xy, xz, and yz planes) using the intersection points found in Step 1: - Intersection with the xy-plane (z=0): The line will pass through the x-axis intersection point \((2, 0, 0)\) and the y-axis intersection point \((0, -3, 0)\). The line equation is given by: $$\frac{y - (-3)}{x - 2} = \frac{-3 - 0}{0 - 2} \implies y = -\frac{3}{2}x + 3$$ - Intersection with the xz-plane (y=0): The line will pass through the x-axis intersection point \((2, 0, 0)\) and the z-axis intersection point \((0, 0, 6)\). The line equation is given by: $$\frac{z - 0}{x - 2} = \frac{6 - 0}{0 - 2} \implies z = -3x + 6$$ - Intersection with the yz-plane (x=0): The line will pass through the y-axis intersection point \((0, -3, 0)\) and the z-axis intersection point \((0, 0, 6)\). The line equation is given by: $$\frac{z - 0}{y - (-3)} = \frac{6 - 0}{0 - (-3)} \implies z = 2y + 6$$
03

Sketch a graph of the plane

Use the intersection points (\((2, 0, 0)\), \((0, -3, 0)\), and \((0, 0, 6)\)) to create a triangular representation of the plane. Draw lines to connect each of the points, resulting in a triangular plane within the 3D coordinate system. Label the points and the intersection lines found in Steps 1 and 2.

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Key Concepts

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

3D coordinate system
The 3D coordinate system is a foundational concept in algebra and geometry that extends the ideas of the 2D coordinate plane (x and y-axes) into three dimensions by adding a z-axis. This z-axis is perpendicular to both the x and y-axes creating a three-dimensional space. In this space, any point can be represented by a set of three coordinates \(x, y, z\).

In our example, we look at where a plane intersects with the coordinate axes in a 3D system. Each axis intersection point represents a key piece of information that helps us visualize the position and orientation of the plane in three-dimensional space. Identifying these intersections is a common exercise in understanding the spatial relationships within the 3D coordinate system.
Equations of lines
In the context of a 3D coordinate system, the equation of a line is the algebraic representation of that line, usually expressed in terms of its variables. Lines can be described by equations in various forms including the slope-intercept form \(y = mx + b\) for 2D or parametric equations and symmetric equations for 3D.

In solving for where a plane intersects coordinate planes in our example, we determine the equations of lines in 3D space. These equations help us describe the exact location and path of the intersection lines along the coordinate planes. The salient point about these lines is that they serve to outline the trace of the plane where it slices through each of the xy, xz, and yz planes, illustrating the complex interplay between linear and planar elements in three dimensions.
Graphing planes
Graphing a plane involves plotting its position in a three-dimensional space. A plane in 3D can be represented by an equation of the form \(Ax + By + Cz + D = 0\), where \(A\), \(B\), and \(C\) are coefficients that influence the orientation of the plane, and \(D\) is a constant that affects the plane's position relative to the origin.

When graphing, the intersection points calculated where the plane meets the x, y, and z-axes serve as critical guides. Connecting these points visualizes the extent of the plane within the coordinate system. The sketch thus created provides a visual understanding of the plane's orientation and is instrumental for spatial comprehension in subjects like solid geometry and vector calculus.
Intersection of planes with axes
The intersection of planes with axes in a 3D coordinate system involves finding the points at which a plane crosses the x, y, and z axes. This is achieved by setting the other two variables to zero and solving for the remaining one.

These intersection points are crucial to understand because they provide the framework to construct the visual representation of the plane by indicating the precise location where the plane 'touches' the axes. Additionally, knowing these points can help find the equation of the lines where the plane intersects with the coordinate planes鈥攕ignificant lines that offer further insight into the nature of the plane within the 3D space.

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Most popular questions from this chapter

Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(\underline{B}\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3} .\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\) d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\) e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0).

Identify and briefly describe the surfaces defined by the following equations. $$-y^{2}-9 z^{2}+x^{2} / 4=1$$

The equation \(x^{2 n}+y^{2 n}+z^{2 n}=1,\) where \(n\) is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a. Find all the extreme points on the flattened sphere with \(n=2\) What is the distance between the extreme points and the origin? b. Find all the extreme points on the flattened sphere for integers \(n>2 .\) What is the distance between the extreme points and the origin? c. Give the location of the extreme points in the limit as \(n \rightarrow \infty\). What is the limiting distance between the extreme points and the origin as \(n \rightarrow \infty ?\)

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\). a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\). c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(\bar{Q}\). d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3. e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?

Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$
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