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Chapter 13: Problem 64

Properties of planes 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. $$12 x-9 y+4 z+72=0$$

Short Answer

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Question: Find the points of intersection of the plane 12x - 9y + 4z + 72 = 0 with the coordinate axes, the equations of the lines where the plane intersects the coordinate planes, and sketch the graph of the plane. Answer: The points of intersection with the coordinate axes are (-6, 0, 0), (0, 8, 0), and (0, 0, -18). The equations of the lines where the plane intersects the coordinate planes are: 1. In the XY plane: \(\frac{y}{x + 6} = \frac{8}{6}\), 2. In the XZ plane: \(\frac{z}{x + 6} = -3\), 3. In the YZ plane: \(\frac{z}{y - 8} = \frac{9}{4}\). To sketch the graph of the plane, plot the intersections with the coordinate axes, connect them to form a triangle, and the resulting graph represents the given plane in 3D space.

Step by step solution

01

Finding the points of intersection with the coordinate axes

Let's set two variables at a time to zero and solve for the third variable. 1. Intersection with the X-axis (y=0, z=0): $$12x - 9(0) + 4(0) + 72 = 0$$ $$12x = -72$$ $$x = -6$$ So, the point of intersection with the X-axis is (-6, 0, 0). 2. Intersection with the Y-axis (x=0, z=0): $$12(0) - 9y + 4(0) + 72 = 0$$ $$-9y = -72$$ $$y = 8$$ So, the point of intersection with the Y-axis is (0, 8, 0). 3. Intersection with the Z-axis (x=0, y=0): $$12(0) - 9(0) + 4z + 72 = 0$$ $$4z = -72$$ $$z = -18$$ So, the point of intersection with the Z-axis is (0, 0, -18). The points of intersection of the plane with the coordinate axes are: (-6, 0, 0), (0, 8, 0), and (0, 0, -18).
02

Finding the equations of the lines where the plane intersects the coordinate planes

1. Intersection with the XY plane (z=0): The intersection points are (-6, 0, 0) and (0, 8, 0). The line passing through these points can be written as: $$\frac{y - 0}{x + 6} = \frac{8 - 0}{0 - (-6)}$$ $$\frac{y}{x + 6} = \frac{8}{6}$$ 2. Intersection with the XZ plane (y=0): The intersection points are (-6, 0, 0) and (0, 0, -18). The line passing through these points can be written as: $$\frac{z - 0}{x + 6} = \frac{(-18) - 0}{0 - (-6)}$$ $$\frac{z}{x + 6} = -3$$ 3. Intersection with the YZ plane (x=0): The intersection points are (0, 8, 0) and (0, 0, -18). The line passing through these points can be written as: $$\frac{z - 0}{y - 8} = \frac{(-18) - 0}{0 - 8}$$ $$\frac{z}{y - 8} = \frac{9}{4}$$ The equations of the lines where the plane intersects the coordinate planes are: 1. In the XY plane: \(\frac{y}{x + 6} = \frac{8}{6}\) 2. In the XZ plane: \(\frac{z}{x + 6} = -3\) 3. In the YZ plane: \(\frac{z}{y - 8} = \frac{9}{4}\).
03

Sketching the graph of the plane

To sketch the graph of the plane, we can plot the points of intersection with the coordinate axes: (-6, 0, 0), (0, 8, 0), and (0, 0, -18). Then, connect these points with straight lines to form a triangle. The lines we found in the previous step are the sides of this triangle in their respective coordinate planes. The resulting graph represents the given plane in 3D space.

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Key Concepts

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

Intersection of Planes with Axes
Understanding the intersection of planes with axes is crucial in visualizing three-dimensional geometry. The process involves finding where a plane, represented by a linear equation such as the given example, cuts through the coordinate axes. To find these points, we typically consider one axis at a time by setting the other two coordinates to zero and solving for the remaining variable.

The textbook solution offered an easy-to-follow method to identify these intersection points by solving simple algebraic equations. For instance, the point on the X-axis is found by setting the y and z values to zero, then solving for x. This technique is a foundational skill in 3D graphing and helps lay the groundwork for deeper exploration of spatial analysis.
Equations of Lines
The concept of equations of lines in coordinate geometry allows us to describe the position and direction of lines using algebraic expressions. In 3D, a line is determined by extending a segment between two points indefinitely in both directions. The steps given to derive the equations where the plane intersects the coordinate planes are a handy application of this principle.

By using two points from the plane's intersection lines with the XY, XZ, and YZ planes, we can formulate linear equations that represent those lines. It's also useful for students to remember the slope-intercept form of a line, and how it can be extended to three dimensions; however, in this particular example, these equations form a simple proportion, derived from the segment's endpoints.
Graphing Planes in 3D
The step to graph the plane in 3D is about converting the algebraic representation of the plane into a visual one. By plotting the points of intersection on a 3D coordinate system and connecting these points, a triangle is formed on the plane. This triangle is integral to understanding the plane's orientation in 3D space.

To make graphing more approachable, it can be helpful to start with plotting each axis-intersecting point. After marking these points, draw straight lines connecting them, which helps visualize the edges of the plane. Students should be encouraged to practice by graphing a few planes to grow comfortable with the transition from equations to visual representations.
Coordinate Axes Intersection Points
Familiarity with coordinate axes intersection points empowers students to understand where a plane cuts through each axis. These points are essential when sketching the plane as they frame its position in space. As detailed in the given solution, finding where a plane intersects the X, Y, and Z axes is a matter of algebraic manipulation. Setting two variables to zero and resolving for the third gives us these points.

When students grasp this concept, they gather crucial insight into the 3D layout of geometric figures and how algebra relates to spatial configurations. Intersection points also serve as a bridge to more advanced topics, such as calculating distances in space and working with vectors and transformations.

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Most popular questions from this chapter

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}=\operatorname{proj}_{\mathbf{u}} \mathbf{v}\) b. If nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) have the same magnitude, they make equal angles with \(\mathbf{u}+\mathbf{v}\) c. \((\mathbf{u} \cdot \mathbf{i})^{2}+(\mathbf{u} \cdot \mathbf{j})^{2}+(\mathbf{u} \cdot \mathbf{k})^{2}=|\mathbf{u}|^{2}\) d. If \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\) and \(\mathbf{v}\) is orthogonal to \(\mathbf{w},\) then \(\mathbf{u}\) is orthogonal to \(\mathbf{w}\) e. The vectors orthogonal to\( \langle 1,1,1\rangle\) lie on the same line. f. If proju \(\mathbf{u}=\mathbf{0},\) then vectors \(\mathbf{u}\) and \(\mathbf{v}\) (both nonzero) are orthogonal.

Quadric surfaces Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y\)-, \(x z\), and \(yz\)-traces, when they exist. c. Identify and sketch a graph of the surface. $$4 x^{2}+y^{2}+\frac{z^{2}}{2}=1$$

Suppose \(\mathbf{v}\) is a nonzero position vector in the \(x y\) -plane. How many position vectors with length 2 in the \(x y\) -plane are orthogonal to \(\mathbf{v}\) ?

Sketching planes Sketch the following planes in the window \([0,5] \times[0,5] \times[0,5]\) $$x=2$$

Vectors with equal projections Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors u with the same value of proj\(_{\mathbf{v}} \mathbf{u} .\) Find another vector that has the same projection onto \(\mathbf{v}=\langle 1,1\rangle\) as \(\mathbf{u}=\langle 1,2\rangle .\) Draw a picture.
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