Problem 7 - University Calculus: Early Transcendentals

Question

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+z^{2}=4, \quad y=0$$

Step-by-Step Solution

Answer

The set of points is a circle of radius 2 in the \(xz\)-plane at \(y = 0\).

Step 1: Understand the Equation Pair

The given pair of equations is: 1. \( x^{2} + z^{2} = 4 \) 2. \( y = 0 \). The first equation, \( x^2 + z^2 = 4 \), represents a circle with radius 2 in the \(xz\)-plane. The second equation, \( y = 0 \), indicates that this shape is constrained to the \(xz\)-plane, meaning no deviation along the \(y\)-axis.

Step 2: Identify the Geometric Shape

The equation \(x^2 + z^2 = 4\) describes a circle of radius 2 centered at the origin in the \(xz\)-plane. The equation \(y = 0\) restricts this circle to lie in the \(xz\)-plane, meaning all points on the circle have \(y\) coordinate as 0.

Step 3: Describe the Geometric Set in Space

Since \(x^2 + z^2 = 4\) forms a circle within the \(xz\)-plane and \(y = 0\) ensures the entire structure stays on this plane, the set of points represented by these equations is a two-dimensional circle. Thus, in three-dimensional space, the circle lies entirely along the \(xz\)-plane, perpendicular to the \(y\)-axis.

Key Concepts

Coordinate System

In the realm of geometry and algebra, a coordinate system is a framework that allows us to describe every point in space with a set of numbers. The most commonly used system in three-dimensional spaces is the Cartesian coordinate system, which uses three axes, usually labeled as the x-axis, y-axis, and z-axis. Each point within this system is defined by an ordered triplet \( (x, y, z) \). This triplet specifies the point's exact location by determining its distance along each of the three axes. * The **x-axis** typically represents the horizontal direction. * The **y-axis** often represents the vertical direction. * The **z-axis** comes out perpendicularly from the x-y plane, adding depth into the three-dimensional space.By using these three axes, any point can be pinpointed with precision. In our scenario, the focus is on the \( xz\) plane due to the equation \( y = 0\), which means all points have no vertical displacement. This highlights how the coordinate system can help isolate specific planes or lines of interest by manipulating axes values.

Equations

Equations are mathematical statements that express the equality between two expressions. They serve as fundamental tools for describing relationships between different quantities. In the context of geometry, equations often describe shapes and sets within a given space.Let's consider the provided equations: * \( x^2 + z^2 = 4 \) * \( y = 0 \)The first equation, \( x^2 + z^2 = 4\), represents a circle with a radius of 2 centered at the origin within the xz-plane. This is derived from the equation of a circle, \( (x-h)^2 + (z-k)^2 = r^2\), where \( (h, k)\) is the center and \( r\) is the radius.The second equation, \( y = 0\), means that the circle does not vary along the y-axis at all. It ensures that every point defined by the circle equation lies flat, directly on the xz-plane.Together, these equations describe a circle with no height, bound entirely to the xz-plane, illustrating the power of algebraic expressions in dictating geometric properties.

3D Space

Understanding 3D space involves comprehending how objects orient themselves in a volume, using three axes for reference. In a three-dimensional coordinate space, every location is described through coordinates \( (x, y, z) \). Unlike two-dimensional space, where only length and width are considered, 3D space includes depth, offering a full understanding of the spatial relationships.In this particular task, we've explored how a simple set of equations can define structures within this volumetric space. The equations \( x^2 + z^2 = 4 \) and \( y = 0 \) define a circle that resides entirely along the xz-plane. This means the circle is set at the intersection of the x and z dimensions, and extends no further into the space. Imagine cutting a flat circle out of paper and laying it on a table; this mirrors how the circle exists in 3D space, constrained on the plane where y remains constant at zero.Such considerations are crucial in 3D modeling, as they allow for the precise placement and understanding of objects in a space that initially seems complex but is inherently ordered and structured through the x, y, and z coordinates.