Problem 4 In Exercises $1-16,$ give a ge... [FREE SOLUTION]

Chapter 12: Problem 4

In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=1, \quad y=0$$

Short Answer

Expert verified

The set of points is a vertical line parallel to the z-axis at x=1 and y=0.

Step by step solution

Interpret the Equations

We start by understanding what each equation represents: the equation $x=1$ describes all points in space that have an x-coordinate of 1. Similarly, the equation $y=0$ describes all points with a y-coordinate of 0.

Find the Intersection of the Equations

Since both conditions must be satisfied simultaneously, we need to find the set of points that have both an x-coordinate of 1 and a y-coordinate of 0. This requires both $x=1$ and $y=0$ to hold at the same time.

Geometric Interpretation

The intersection of the planes $x=1$ and $y=0$ is a line. Specifically, it is a vertical line in space parallel to the z-axis, passing through the point $(1, 0, 0)$. This line extends infinitely in the positive and negative z-directions.

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Key Concepts

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

Intersection of planes

When we talk about the intersection of planes in a three-dimensional space, we are referring to the geometric line where two planes meet. Imagine two sheets of paper (planes) crossing each other. Where they overlap, a straight line is formed. This is common in geometry, where planes can intersect in a line or not intersect at all if they are parallel.

The intersection can be a helpful visual tool to understand spatial relationships between different planes.
When two planes intersect at a line, every point on that line is common to both planes.

In our exercise, the equations of the planes are given by the conditions:
- Plane 1: All points where $x = 1$.
- Plane 2: All points where $y = 0$.
These planes intersect to form a line in space.

Coordinate system

A coordinate system in mathematics, especially in geometry, is a method for identifying each point uniquely in space through coordinates. The most common system is the Cartesian coordinate system which uses x, y, and z axes for identifying locations in a three-dimensional space. This could be compared to assigning an address to each point in the space.

Each point has a unique combination of values (x, y, z).
These coordinates help in understanding and analyzing geometrical concepts.

Using a coordinate system allows us to clearly define both the position and direction of a line or plane.
In our specific task, the point with coordinates $(1, 0, z)$ is used to describe the position along the line of intersection.

Line in space

A line in space is a straight one-dimensional figure that extends infinitely in both directions in a three-dimensional space. When two planes intersect, the result is typically a line. The line can be described using linear equations, which clarify how the points align in space. This line will have points that comply with both planes' conditions.

Lines in space offer a way to understand the concept of slope and directionality beyond two dimensions.
These lines can be defined parametrically or through their directional vector.

For our example, the intersection is defined by both $x = 1$ and $y = 0$.
This results in a line parallel to the z-axis, making it vertical in space and consistent at every z value.

Equations of planes

The equation of a plane in a three-dimensional coordinate system is typically expressed as $ax + by + cz = d$. Each of these terms represents a dimension, and each coefficient reflects the orientation of the plane.

The coefficients $a$, $b$, and $c$ determine the plane's tilt, while $d$ adjusts its position in space.
Simpler equations can define special types of planes, such as those parallel to the coordinate planes.

In the exercise at hand, the equations $x = 1$ and $y = 0$ describe planes which are parallel to two of the three major axes.
This makes the analysis of their intersection straightforward, leading to a line that extends indefinitely along the z-axis.

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91影视

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=1, \quad y=0$$

Short Answer

Step by step solution

Interpret the Equations

Find the Intersection of the Equations

Geometric Interpretation

Key Concepts

Intersection of planes

Coordinate system

Line in space

Equations of planes

One App. One Place for Learning.

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