Problem 16 - Advanced Engineering Mathematics

Question

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ x^{2}+y^{2}+z^{2}=0 $$

Step-by-Step Solution

Answer

The locus of points is the origin, (0, 0, 0).

Step 1: Understanding the Equation

The equation given is a sphere's equation in three-dimensional space: \(x^2 + y^2 + z^2 = 0\). A typical sphere equation is \(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius and \((0,0,0)\) is the center of the sphere. In this case, \(r^2 = 0\).

Step 2: Analyzing the Radius

Since \(r^2 = 0\), the radius \(r\) is 0. A sphere with a zero radius means that all the points satisfying the equation \(x^2 + y^2 + z^2 = 0\) must lie on a single point.

Step 3: Identifying the Point

If \(x^2 + y^2 + z^2 = 0\), each of the terms \(x^2\), \(y^2\), and \(z^2\) must also be zero because they are squares of real numbers. Thus, \(x=0\), \(y=0\), and \(z=0\).

Step 4: Describing the Locus

The locus of points \(P(x, y, z)\) that satisfy \(x^2 + y^2 + z^2 = 0\) is the single point (0, 0, 0). This point is often referred to as the origin in three-dimensional space.

Key Concepts

Three-Dimensional Space

Three-dimensional space is the world we inhabit, consisting of three axes that define a location: the x-axis, y-axis, and z-axis. This three-dimensional coordinate system helps us locate points in space using a triplet of values \(x, y, z\). Each axis represents a dimension: \(x\) for width, \(y\) for height, and \(z\) for depth.

In math, especially when dealing with geometry and algebra, this space allows us to express points, vectors, shapes, and lines in a more comprehensive manner compared to two-dimensional space.

• Points in three-dimensional space can be described using coordinates like \( (x, y, z) \).
• Lines and planes can also be defined through equations involving these three variables.
• The concept of distance and location in this space combines these three dimensions.

Understanding this space is crucial when exploring geometric objects, such as spheres, cubes, or any object that occupies a volume rather than just an area.

Sphere Equation

The general equation of a sphere in three-dimensional space is \(x^2 + y^2 + z^2 = r^2\). This formula represents a sphere centered at the origin \( (0, 0, 0) \) with a radius \(r\). A sphere can be thought of as all points in space that are equidistant from a central point.

In the exercise, the sphere equation simplifies to \(x^2 + y^2 + z^2 = 0\), indicating a special case. When the radius \(r\) becomes zero, it implies that the sphere collapses into a single point, which is the center of the sphere itself.

• In this simplified sphere equation, the locus of points is reduced to the point \( (0, 0, 0) \), instead of an expanding surface of a normal sphere.
• Mathematically, each squared term \(x^2, y^2, z^2\) must be zero to satisfy the equation, necessitating that \(x = 0\), \(y = 0\), and \(z = 0\).

This concept highlights a unique instance within spheres, where shrinking the radius to zero results in a nonexistent volume but points to a crucial point in space.

Origin Point

The origin point in three-dimensional space is represented by coordinates \( (0,0,0) \). It serves as the central reference point for this space and lies at the intersection of the x, y, and z axes.

In mathematics, particularly in algebra and geometry, the origin is pivotal in analyzing and interpreting three-dimensional objects and their properties.

• It acts as the fixed zero-point from which other points or vectors can be measured.
• The origin is crucial for defining symmetries, rotations, and transformations in space.
• In our exercise, the locus described by the sphere equation \(x^2 + y^2 + z^2 = 0\) identifies precisely the origin point, illustrating its significance as the anchor of this geometrical representation.

Understanding the role of the origin allows us to explore more complex shapes and movements within three-dimensional systems, making it a fundamental part of spatial comprehension.