Problem 28 - Thomas Calculus

Question

In Exercises \(25-34\) , describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at \((0,0,0)\) and lying in the $$ \text { a. } x y \text { -plane } \quad \text { b. } y z \text { -plane } \quad \text { c. } x z $$

Step-by-Step Solution

Answer

a. \(x^2 + y^2 = 4\), b. \(y^2 + z^2 = 4\), c. \(x^2 + z^2 = 4\).

Step 1: Understanding the Circle Equation in 3D

The equation of a circle in 3D space is determined by its center and radius. The general equation for a circle in the standard form is \((x - h)^2 + (y - k)^2 = r^2\) if the circle lies in the XY plane. If the circle lies in the XYZ space, we have to factor in the constraint from the plane it lies on.

Step 2a: Circle in the XY-plane

For a circle in the XY-plane centered at the origin \((0,0,0)\) with a radius of 2, the equation simplifies to \(x^2 + y^2 = 4\). This equation describes a circle lying flat in the XY plane, with the Z coordinate constantly 0.

Step 2b: Circle in the YZ-plane

For a circle in the YZ-plane centered at the origin \((0,0,0)\) with a radius of 2, the equation simplifies to \(y^2 + z^2 = 4\). This circle is lying flat in the YZ plane with the X coordinate constantly 0.

Step 2c: Circle in the XZ-plane

For a circle in the XZ-plane centered at the origin \((0,0,0)\) with a radius of 2, the equation simplifies to \(x^2 + z^2 = 4\). This circle is lying flat in the XZ plane with the Y coordinate constantly 0.

Key Concepts

Circle Equations

A circle in mathematics is defined by its center and radius. In two-dimensional space, its equation is usually written as - \((x - h)^2 + (y - k)^2 = r^2\), - where - \((h, k)\) is the center of the circle, - \(r\) is the radius. - Visualizing this in three-dimensional space introduces complexity, as the position of the circle is now related to the coordinate plane in which it lies. For example, if a circle of radius 2 is centered at the origin - \((0,0,0)\) - and lies in the XY-plane, its equation remains as - \(x^2 + y^2 = 4\). - This occurs because the - Z-coordinate is consistently zero, demonstrating how circles can be aligned differently within different planes yet maintain the same fundamental equation structure.

Coordinate Planes

Coordinate planes are integral in 3D geometry as they provide a foundation to locate objects like points, lines, and circles. In three-dimensional space, we primarily deal with three coordinate planes: - XY-plane, - YZ-plane, and - XZ-plane. Each plane serves as a two-dimensional flat surface slicing through the 3D space. Here's how they work: - The **XY-plane** is where - Z is always zero, - around which the - x and y - coordinates can change freely. - The **YZ-plane** fixes - x at zero, - allowing only - y - and - z - values to vary. - Finally, the **XZ-plane** is where - y remains zero - with - x and - z - changing values. - Familiarity with these planes is crucial for understanding how geometric figures like circles are oriented in space.

Three-Dimensional Space

Understanding three-dimensional space is foundational for tackling more complex geometrical concepts. In 3D geometry, each point is defined by a set of coordinates - ":[x, y, z]", - which helps precisely position objects in space. This contrasts with 2D geometry that only requires - ":[x, y]". 3D space allows us to explore and analyze objects that extend not just in height and width but also in depth. Key aspects include: - **Points:** Defined by their three coordinates, - represent positions. - **Lines and Curves:** Connections between points or paths they travel. - **Planes:** Flat surfaces slicing through the space, crucial for orientation. - **Shapes and Solids:** Viewable from multiple perspectives and not restricted to flat images. By grasping these elements, students can better comprehend how circles and other figures integrate and interact within a three-dimensional framework, enabling ease of understanding in exercises involving mathematical modeling of such shapes.