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Chapter 11: Problem 104

Describe each surface given by the equations \(x=a\), \(y=b\), and \(z=c\).

Short Answer

Expert verified
The equations \(x=a\), \(y=b\), and \(z=c\) represent planes in 3D space. The plane \(x=a\) is parallel to the yz-plane and is 'a' units away from it along the x-axis. The plane \(y=b\) is parallel to the xz-plane and is 'b' units away from it along the y-axis. The plane \(z=c\) is parallel to the xy-plane and is 'c' units away from it along the z-axis.

Step by step solution

01

Understand what \(x=a\) represents

The equation \(x=a\) is a vertical plane that is parallel to the yz-plane. The plane is a distance of 'a' units away from the yz-plane along the x-axis. The value of 'a' moves the plane along the x-axis, meaning if 'a' were to increase, the plane would shift rightwards.
02

Understand what \(y=b\) represents

The equation \(y=b\) is a vertical plane that is parallel to the xz-plane. This plane is a distance of 'b' units away from the xz-plane along the y-axis. The value of 'b' moves the plane along the y-axis, meaning if 'b' were to increase, the plane would shift upwards.
03

Understand what \(z=c\) represents

The equation \(z=c\) represents a horizontal plane parallel to the xy-plane. This plane is a distance of 'c' units away from the origin along the z-axis. The value of 'c' moves the plane along the z-axis, meaning if 'c' were to increase, the plane would shift upwards.

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Key Concepts

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

Planes
In analytical geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. Unlike lines, planes need two dimensions within them to be completely described. In three-dimensional space, a plane can be defined using an equation, generally in the form of
  • Equation: Like the ones we've seen, such as \(x=a\), \(y=b\), or \(z=c\). Each equation represents a plane that is parallel to one of the coordinate planes (such as yz-plane, xz-plane, and xy-plane, respectively).
  • Properties:
    • A plane doesn't have thickness.
    • It intersects with three-dimensional space to form a line where another plane or line intersects it.
Imagine slicing a cake with a very thin knife; the flat surface created by the knife cut represents a plane. In our example:
  • The equation \(x=a\) creates a plane parallel to the yz-plane, shifted by 'a' units along the x-axis.
  • The equation \(y=b\) forms a plane parallel to the xz-plane, moved 'b' units along the y-axis.
  • Finally, \(z=c\) defines a plane parallel to the xy-plane, displaced 'c' units along the z-axis.
Each change in 'a', 'b', or 'c' affects the placement of the plane within three-dimensional space, shifting it along the respective axis.
Coordinate System
A coordinate system is a framework used to precisely describe the location of points in space. Understanding coordinate systems is crucial when working with planes and analyzing their positions in three-dimensional space. In a three-dimensional coordinate system, there are three perpendicular axes:
  • x-axis: Runs horizontally, determining the position of points along a left-to-right direction. It's usually represented in equations as 'x'.
  • y-axis: Runs vertically, determining the position of points in an up-and-down fashion. Represented as 'y' in equations.
  • z-axis: Extends depth-wise, often perceived as the in-and-out movement from the observer's perspective. Represented as 'z'.
Using these axes, we can pinpoint exact locations by using an ordered triplet of numbers (x, y, z). For example, the point (3, 5, 2) would be 3 units along the x-axis, 5 units along the y-axis, and 2 units along the z-axis, creating a unique point in space where these planes or surfaces might intersect or be situated.
Three-dimensional Space
Three-dimensional space, often referred to as 3D space, is where all points have a unique set of coordinates composed of three values: x, y, and z. It allows us to visualize and describe objects and their relationships within physical environments.In this context:
  • Dimensions: Our world is three-dimensional because it allows depth (z), in addition to height (y) and width (x).
  • Positioning: Every object or point in 3D space can be located using coordinates (x, y, z), which is why understanding each axis is vital.
  • Applications: 3D space is fundamental not only in geometry but also in fields like physics, computer graphics, and engineering, where the position and interaction of elements need precise calculations.
Concepts such as planes stem from the understanding of 3D space as they "cut" through this space, creating visual and analytical interpretations of how objects might sit relative to one another. When working through problems like the one with the equations \(x=a\), \(y=b\), and \(z=c\), understanding these fundamentals of 3D space makes it easier to grasp how surfaces interact within it.

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Most popular questions from this chapter

Write an equation whose graph consists of the set of points \(P(x, y, z)\) that are twice as far from \(A(0,-1,1)\) as from \(B(1,2,0)\)

\mathrm{\\{} M o d e l i n g ~ D a t a ~ P e r ~ c a p i t a ~ c o n s u m p t i o n s ~ ( i n ~ g a l l o n s ) ~ o f ~ different types of plain milk in the United States from 1994 to 2000 are shown in the table. Consumption of light and skim milks, reduced-fat milk, and whole milk are represented by the variables \(x, y\), and \(z\), respectively. (Source: U.S. Department of Agriculture)$$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 \\ \hline x & 5.8 & 6.2 & 6.4 & 6.6 & 6.5 & 6.3 & 6.1 \\ \hline \boldsymbol{y} & 8.7 & 8.2 & 8.0 & 7.7 & 7.4 & 7.3 & 7.1 \\ \hline z & 8.8 & 8.4 & 8.4 & 8.2 & 7.8 & 7.9 & 7.8 \\ \hline \end{array} $$A model for the data is given by \(0.04 x-0.64 y+z=3.4\) (a) Complete a fourth row in the table using the model to approximate \(z\) for the given values of \(x\) and \(y\). Compare the approximations with the actual values of \(z\). (b) According to this model, any increases in consumption of two types of milk will have what effect on the consumption of the third type?

Show that the plane \(2 x-y-3 z=4\) is parallel to the line \(x=-2+2 t, y=-1+4 t, z=4\), and find the distance between them.

Find a unit vector (a) in the direction of \(\mathrm{u}\) and (b) in the direction opposite of \(\mathbf{u}\). \(\mathbf{u}=\langle 6,0,8\rangle\)

If two lines \(L_{1}\) and \(L_{2}\) are parallel to a plane \(P\), then \(L_{1}\) and \(L_{2}\) are parallel.
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