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Chapter 12: Problem 1

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=2, \quad y=3$$

Short Answer

Expert verified
A line parallel to the z-axis passing through (2, 3) in the xy-plane.

Step by step solution

01

Understand the Equation for x

The equation \(x=2\) specifies that the set of points in space has an x-coordinate of 2. This condition confines all points to a vertical plane parallel to the yz-plane, located at x = 2 in the coordinate system.
02

Understand the Equation for y

The equation \(y=3\) specifies that the set of points in space has a y-coordinate of 3. This condition confines all points to a horizontal plane parallel to the xz-plane, located at y = 3 in the coordinate system.
03

Combine Both Conditions

Combining both equations \(x=2\) and \(y=3\), we need points that satisfy both conditions simultaneously. This results in the intersection of the two planes: the vertical plane \(x=2\) and the horizontal plane \(y=3\).
04

Describe the Intersection Geometrically

Geometrically, the intersection of the vertical plane \(x=2\) and the horizontal plane \(y=3\) is a line. This line is parallel to the z-axis and passes through points (2, 3, z) where z can be any real number. This represents a line in 3D space parallel to the z-axis.

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Key Concepts

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

Planes in 3D space
In the realm of 3D coordinate geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions within its own dimensions. It can be defined by a linear equation, such as in the case of the exercise given:
  • The equation \(x = 2\) describes a vertical plane. Since it is parallel to the yz-plane, it captures all points in space where the x-coordinate equals 2.
  • Similarly, \(y = 3\) describes a horizontal plane that is parallel to the xz-plane, capturing all points where the y-coordinate is 3.
Planes are fundamental in 3D geometry, as they help in understanding spatial relationships. By examining how different planes intersect, one can define lines, points, or even volumes in space. When studying these planes, remember they are not limited; they stretch on endlessly. This concept is crucial as it helps in visualizing and solving more complex geometric problems.
Line intersection
When two planes intersect, as in our problem, they meet along a common line. The intersection of two planes in 3D space is determined by the points that simultaneously satisfy the equations of both planes.
  • In our example, the planes given by \(x = 2\) and \(y = 3\) do not overlap beyond a single line because these conditions restrict the possible positions in space where these planes can meet.
  • The intersection line will include points where both \(x = 2\) and \(y = 3\), with the z-coordinate being free to take any value.
This intersection is visualized as a line running parallel to the z-axis. Understanding this helps in visualizing geometric problems and in recognizing how various elements in a coordinate system can orient and influence one another.
Geometric description of points
In geometric terms, a set of points defined by certain conditions can represent a variety of elements such as lines, planes, or even specific positions. The provided equations \(x = 2\) and \(y = 3\) point to a unique set of conditions.
  • The x-coordinate being fixed at 2 confines these points to a plane where x never changes.
  • The y-coordinate fixed at 3 places these points on another plane where y is constant.
Taken together, these conditions yield a line of points with coordinates \((2, 3, z)\), where \(z\) is an unrestricted variable. This means anyone of these points is defined by limitless potential values of \(z\), forming an infinite line. By grasping this concept, one can understand how to work with complex spatial problems, envisioning how intersections of conditions result in definable geometric figures within 3D spaces.

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