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Chapter 9: Problem 19

In Exercises \(15-20,\) find an equation in rectangular coordinates for the equation given in cylindrical coordinates, and sketch its graph. $$ r^{2}+z^{2}=4 $$

Short Answer

Expert verified
The equivalent equation in rectangular coordinates is \( x^2 + y^2 + z^2 = 4 \), which represents a sphere with the centre at the origin and radius 2.

Step by step solution

01

Understanding Cylindrical and Rectangular Coordinate systems

The rectangular coordinate system is based on x, y and z coordinates while the cylindrical coordinate system uses r, 胃 and z where r is the radius in circular base (which is formed by x and y in rectangular), 胃 is the angle formed by x-axis and line from origin to the point in base and z is height or depth from this base.
02

Converting the given cylindrical coordinates to rectangular coordinates

The given equation is \( r^2 + z^2 = 4 \). In the cylindrical coordinate system, the radius r in the circular base is given by \( r = \sqrt{x^2 + y^2} \). By substituting this into the given equation, we get \( (\sqrt{x^2 + y^2})^2 + z^2 = 4 \). Simplifying this gives \( x^2 + y^2 + z^2 = 4 \).
03

Identifying the graph of the equation

This equation represents the equation of a sphere in three dimensions. The sphere has a center at the origin (0,0,0) and has a radius of 2, since 4 is the square of the radius of the sphere. Therefore, the graph would comprise of a sphere centered at origin and having radius equal to 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Coordinates
Rectangular coordinates, often called Cartesian coordinates, form the basis for a three-dimensional graphing system. They use three values: \(x\), \(y\), and \(z\), which represent length along the respective axes.

These coordinates help locate points in 3D space by using a combination of these values:
  • The \(x\)-coordinate determines the position along the horizontal axis.
  • The \(y\)-coordinate identifies the position along the vertical axis.
  • The \(z\)-coordinate indicates height or depth from the \(xy\)-plane.
All three coordinates combine to describe the exact location of a point in three-dimensional space, making them essential in fields like engineering, physics, and computer graphics. Rectangular coordinates are straightforward and widely used due to their simplicity in representing complex concepts.
Equation Conversion
Equation conversion is an essential concept when transitioning from one coordinate system to another. In this context, it often involves switching from cylindrical to rectangular coordinates.

While converting:
  • The radius \(r\) in cylindrical coordinates transforms to \(\sqrt{x^2 + y^2}\) in rectangular terms, connecting the relationship between these coordinates.
  • Values such as \(\theta\) are not directly visible in basic three-variable expressions; thus, the focus usually remains on translating radial distances and altitudes into \(x, y, z\) components.
This makes it possible to express shapes and geometries described in one system within the other, facilitating better analysis and visualization. By converting equations, we can switch seamlessly between different perspectives of the same geometric entity.
Sphere Equation
A sphere is a classic geometric shape in three-dimensional space and is easily represented in both cylindrical and rectangular coordinates. The equation of a sphere in rectangular coordinates is given by:\[x^2 + y^2 + z^2 = r^2\]where \(r\) is the radius of the sphere.

In our exercise, the equation \( x^2 + y^2 + z^2 = 4 \) describes a sphere centered at the origin with a radius of 2, because 4 is the result of squaring the radius.
  • The center of the sphere at the origin is noted as \((0, 0, 0)\).
  • The radius is derived by recognizing that \(r^2 = 4\), leading to \(r = 2\).
Understanding how these components fit into the sphere's equation helps discern not only the location and scale of the sphere but also its characteristics within its geometric space.
3D Graphing
3D graphing is a powerful tool that allows for the visualization and analysis of equations in three-dimensional space.

When graphing three-dimensional objects like spheres:
  • Tools like 3D modeling software or graphing calculators help in visualizing equations.
  • Axes need to be labeled properly to distinguish \(x\), \(y\), and \(z\).
  • Graphs can represent physical models or concepts that have 3D implications.
In the given exercise, depicting the sphere accurately can reveal insights about symmetry and size in its spatial arrangement.
By plotting such equations, students gain an understanding of both the mathematical and tangible nature of three-dimensional objects. Mastering 3D graphing enhances skills in spatial reasoning and further applications in engineering, architecture, and natural sciences.

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Most popular questions from this chapter

Prove the triangle inequality \(\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|\).

In Exercises 45 and \(46,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } \mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w} \text { and } \mathbf{u} \neq \mathbf{0}, \text { then } \mathbf{v}=\mathbf{w} $$

Find the magnitude of \(v\). Initial point of \(\mathbf{v}:(1,-3,4)\) Terminal point of \(\mathbf{v}:(1,0,-1)\)

(a) find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and (b) find the vector component of u orthogonal to v. $$ \mathbf{u}=\langle 1,0,4\rangle, \quad \mathbf{v}=\langle 3,0,2\rangle $$

Use vectors to determine whether the points are collinear. (0,0,0),(1,3,-2),(2,-6,4)
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