Problem 4

Question

Describe in words the set of all points in $\mathbf{R}^{3}$ that satisfy the following expressions: (a) $x=0$ (b) $y=0$ (c) $z=0$ (d) $z \geq 0$ (e) $y \leq 0$

Step-by-Step Solution

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Answer

Points on yz-plane, xz-plane, xy-plane, half-space above xy-plane, and half-space below xz-plane.

1Step 1: Understanding Expression (a)

The expression $x = 0$ describes all points in three-dimensional space where the x-coordinate is zero. This means that the points lie on the yz-plane. The yz-plane is the plane parallel to the x-axis that contains the y-axis and z-axis.

2Step 2: Understanding Expression (b)

The expression $y = 0$ describes all points where the y-coordinate is zero. These points lie on the xz-plane. The xz-plane contains all points that are parallel to the y-axis and lies in the direction of the x and z axes.

3Step 3: Understanding Expression (c)

The expression $z = 0$ describes all points where the z-coordinate is zero. These points lie on the xy-plane. The xy-plane contains all x and y points and is parallel to the z-axis.

4Step 4: Understanding Expression (d)

The expression $z \geq 0$ defines all points in space where the z-coordinate is greater than or equal to zero. This includes the entire half-space above and including the xy-plane.

5Step 5: Understanding Expression (e)

The expression $y \leq 0$ describes the set of points where the y-coordinate is less than or equal to zero. This includes the entire half-space on or below the xz-plane.

Key Concepts

Understanding the yz-planeUnderstanding the xz-planeUnderstanding the xy-planeUnderstanding half-space in 3D

Understanding the yz-plane

In three-dimensional space, when we talk about the yz-plane, we are referring to a very specific set of points. These are all the points where the x-coordinate is equal to zero. Imagine it as a flat sheet that stretches across the y and z axes, leaving out the x-axis. You can think of it like a page in a book that lies between the two book covers (in this case, the y-axis and z-axis), standing upright without touching the x-axis.

The yz-plane is parallel to the x-axis.
It includes points such as (0, 1, 2), (0, -3, 5), and (0, 4, -2).
For any point on this plane, the equation is defined by setting the x-coordinate to zero: $x = 0$.
This plane is crucial for graphing and visualizing concepts in 3D space.

Understanding the xz-plane

The xz-plane is another major reference plane in three-dimensional space. This plane is all about the points where the y-coordinate is zero. Picture it as the flat surface spanning between the x-axis and the z-axis like a perfectly horizontal sheet on a table.

The xz-plane runs parallel to the y-axis.
Examples of points on the xz-plane are (1, 0, 2), (-3, 0, 5), and (4, 0, -2).
To find the xz-plane, set the y-coordinate to zero: $y = 0$.
Understanding this plane helps in visualizing changes happening in x and z directions only.

Understanding the xy-plane

The xy-plane is a fundamental plane in 3D space that includes points where the z-coordinate is zero. Imagine it as a big flat stage that the axes x and y create, extending infinitely in those directions.

The xy-plane is parallel to the z-axis.
Points like (1, 2, 0), (-3, 5, 0), and (4, -2, 0) are all on the xy-plane.
The equation for this plane is expressed by $z = 0$.
This plane is widely used in mathematical modeling and drawing projections.

Understanding half-space in 3D

In three-dimensional geometry, a half-space is one part of space divided by a plane. You can think of it as one half of a sandwich where a plane acts like the bread dividing it.

Positive and negative half-space:

The expression $z \geq 0$ defines a positive half-space above the xy-plane, encompassing all points where the z-coordinate is zero or positive. It includes places like (1, 2, 0) or (-3, 5, 4).
Similarly, $y \leq 0$ forms a negative half-space along or below the xz-plane, covering points such as (4, 0, -1) or (-2, -3, -4).

Significance of half-spaces:

Half-spaces are useful in inequalities and optimization problems in mathematics.
Dividing spaces into half-spaces helps in analyzing and describing geometric locations.

Problem 4

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