Problem 16 - Precalculus with Calculus Previews

Question

Describe the set of points \(P(x, y,\)z) in 3-space whose coordinates satisfy the given equation. $$ x^{2}+y^{2}+z^{2}=0 $$

Step-by-Step Solution

Answer

The set of points is a single point: (0, 0, 0).

Step 1: Analyze the Equation

The equation given is \( x^2 + y^2 + z^2 = 0 \). This is an expression that represents a set of points \((x, y, z)\) in 3-space.

Step 2: Consider Each Square Term

Notice that \( x^2, y^2, \) and \( z^2 \) are all squared terms, which are always non-negative (i.e., \( x^2 \geq 0, y^2 \geq 0, \) and \( z^2 \geq 0 \)). The sum of non-negative numbers \( x^2 + y^2 + z^2 = 0 \) implies that each must individually be zero, because they cannot negate each other or become negative.

Step 3: Solve for Each Variable

Since \( x^2 = 0 \), \( y^2 = 0 \), and \( z^2 = 0 \), the only solution for each is \( x = 0 \), \( y = 0 \), and \( z = 0 \). Thus, the coordinates \((x, y, z)\) must all be zero.

Step 4: Describe the Point Set in 3-Space

The set of points satisfying the given equation \( x^2 + y^2 + z^2 = 0 \) consists of only a single point, which is the origin \((0, 0, 0)\) in the 3D space.

Key Concepts

Equation of a Sphere

In 3-dimensional coordinate geometry, an equation of a sphere is typically written as \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]where

• \((h, k, l)\) represents the center of the sphere.
• \(r\) stands for the radius of the sphere.

For the given equation \(x^2 + y^2 + z^2 = 0\), you might wonder where the center is and what the radius could be.
The equation here is a special case where all values are squared and summed to equal zero.
Considering basic algebra, \[x^2 = 0, \, y^2 = 0, \, z^2 = 0\]means that

• \(x = 0\)
• \(y = 0\)
• \(z = 0\)

With these values, it turns out that this equation does not represent a sphere with a positive radius. Instead, it describes something that exists only at the central point, the origin. This is because a radius cannot be zero and still define a sphere with volume.
Therefore, this specific equation is centered at the origin with a radius of zero, identifying just a single point!

Set of Points in Space

When discussing the set of points in 3-space, we often refer to all possible positions where points can exist based on given conditions.
In geometry, each point is defined by an ordered triple \((x, y, z)\).
In our specific exercise, the equation \(x^2 + y^2 + z^2 = 0\) describes a set of points.
Let's break this down:
Normally, we expect an infinite set of points forming a shape, like a line or surface. However, the special requirement here, having each squared distance equal to zero, leads to a unique scenario.

• Since a square term cannot be negative, the only possible solution is where all squared terms equal zero simultaneously.
• This condition leads straight to the single point \((0, 0, 0)\) as every coordinate must independently satisfy zero.

Thus, the set of points described isn't spread across space but instead is collapsed into just one point
— right at the origin of the coordinate system. For this reason, while the general term 'set' implies multiple elements, in this exercise, the 'set' is effectively one single point.

Origin in 3-Space

The origin in 3-dimensional space is a fundamental concept acting as the central reference point. Notated as \((0, 0, 0)\), the origin is where the three coordinate planes intersect. This point is crucial in geometry as it serves as the fixed center of the coordinate system and is the starting point for measuring distances in any direction.
In the case of the given equation \(x^2 + y^2 + z^2 = 0\), the origin emerges as the sole solution. Here are some key points to understand:

• Each coordinate \(x, y, z\) is zero at the origin, which aligns perfectly with the equation's requirements.
• Since no variability or movement exists around zero for each term, no other points can satisfy the equation.
• The origin is sometimes represented visually as a simple point in space, but conceptually, it marks the convergence of all axes: \(x\)-, \(y\)-, and \(z\)-.

The origin can be considered the grounding point for 3D calculations, providing the balance and centroid for the coordinate system. Therefore, in this context, when all terms zero out, the origin becomes the central, solitary expression of the equation, a reminder of its significance and stability in spatial geometry.