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Chapter 11: Problem 75

Use a computer algebra system or graphing utility to convert the point from one system to another among the rectangular, cylindrical, and spherical coordinate systems. \((5, \pi / 9,8)\)

Short Answer

Expert verified
The coordinates in the rectangular system are \((5 \cdot \cos(\pi / 9), 5 \cdot \sin(\pi / 9), 8)\), and in the spherical system are \((\sqrt{89}, \arctan(\frac{5}{8}), \pi / 9)\)

Step by step solution

01

Conversion to Rectangular Coordinates

In a cylindrical coordinate system, the conversion to the rectangular or Cartesian coordinate system (x, y, z) is carried out using the following transformations: \(x = r \cdot \cos \theta\), \(y = r \cdot \sin \theta\), \(z = z\). Substituting the given cylindrical coordinates \((5, \pi / 9,8)\), we get \(x = 5 \cdot \cos(\pi / 9)\), \(y = 5 \cdot \sin(\pi / 9)\), and \(z = 8\). Hence, the point in rectangular coordinates is \((5 \cdot \cos(\pi / 9), 5 \cdot \sin(\pi / 9), 8)\)
02

Conversion to Spherical Coordinates

To convert cylindrical coordinates to spherical coordinates (蟻, 蠁, 胃), we use the formulas: \(蟻 = \sqrt{x^2 + y^2 + z^2} = \sqrt{r^2+z^2}\), \(\phi = \arctan(\frac{r}{z})\), and \(胃 = 胃\). By substituting \(r = 5\), \(z = 8\), and \(胃 = \pi / 9\), we get \(蟻 = \sqrt{5^2 + 8^2}\), \(\phi = \arctan(\frac{5}{8})\), and \(胃 = \pi / 9\). Hence, the point in spherical coordinates is \((\sqrt{89}, \arctan(\frac{5}{8}), \pi / 9)\)

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Key Concepts

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

Cylindrical Coordinates
Cylindrical coordinates offer a way to specify the location of a point in three-dimensional space, similar to the polar coordinate system used in two dimensions, but with an extra dimension for height. In cylindrical coordinates, a point is expressed as
  • \((r, \theta, z)\) - \(r\) is the distance from the origin to the point's projection on the xy-plane.
  • \(\theta\) is the angle formed with the positive x-axis, measured in the xy-plane.
  • \(z\) is the height above the xy-plane.
To convert from cylindrical to rectangular coordinates:
  • x is calculated by \(x = r \cdot \cos \theta\).
  • y is determined by \(y = r \cdot \sin \theta\).
  • z remains unchanged.
These transformations translate a cylindrical point like \((5, \pi / 9, 8)\) into its rectangular equivalent using the given formulas.
Rectangular Coordinates
The rectangular coordinate system, also known as Cartesian coordinates, is one of the most familiar systems for locating points in space. Each point is defined by three values:
  • \(x\), which indicates horizontal displacement.
  • \(y\), showing vertical displacement along the perpendicular horizontal axis.
  • \(z\), representing elevation above the xy-plane.
This system excels in simplicity because it uses perpendicular directions, making it intuitive for graphing on paper or digital platforms.
For instance, given cylindrical coordinates \((5, \pi / 9, 8)\), transformation results yield rectangular coordinates as \((5 \cdot \cos(\pi / 9), 5 \cdot \sin(\pi / 9), 8)\).
This direct clear layout allows for a straightforward understanding of spatial positions and is ideal for operations and visualizations in engineering and physics.
Spherical Coordinates
Spherical coordinates are perfect when working with points confined to a sphere or needing radial symmetry. This system describes a point in space with three components:
  • \(\rho\), the radial distance from the origin to the point.
  • \(\phi\), the angle formed with the positive z-axis, often called the polar angle or colatitude.
  • \(\theta\), the azimuthal angle, identical to that in cylindrical coordinates.
To convert cylindrical coordinates \((r, \theta, z)\) to spherical coordinates, we use:
  • \(\rho = \sqrt{r^2 + z^2}\).
  • \(\phi = \arctan\left(\frac{r}{z}\right)\).
  • \(\theta = \theta\).
Using these formulas, the point \((5, \pi / 9, 8)\) in cylindrical coordinates becomes \((\sqrt{89}, \arctan(\frac{5}{8}), \pi / 9)\) in spherical coordinates, visually making it easier to represent volumetric and radial data.

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Most popular questions from this chapter

\mathrm{\\{} M o d e l i n g ~ D a t a ~ P e r ~ c a p i t a ~ c o n s u m p t i o n s ~ ( i n ~ g a l l o n s ) ~ o f ~ different types of plain milk in the United States from 1994 to 2000 are shown in the table. Consumption of light and skim milks, reduced-fat milk, and whole milk are represented by the variables \(x, y\), and \(z\), respectively. (Source: U.S. Department of Agriculture)$$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 \\ \hline x & 5.8 & 6.2 & 6.4 & 6.6 & 6.5 & 6.3 & 6.1 \\ \hline \boldsymbol{y} & 8.7 & 8.2 & 8.0 & 7.7 & 7.4 & 7.3 & 7.1 \\ \hline z & 8.8 & 8.4 & 8.4 & 8.2 & 7.8 & 7.9 & 7.8 \\ \hline \end{array} $$A model for the data is given by \(0.04 x-0.64 y+z=3.4\) (a) Complete a fourth row in the table using the model to approximate \(z\) for the given values of \(x\) and \(y\). Compare the approximations with the actual values of \(z\). (b) According to this model, any increases in consumption of two types of milk will have what effect on the consumption of the third type?

Let \(\mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{j}+\mathbf{k}\), and \(\mathbf{w}=\overline{a \mathbf{u}}+b \mathbf{v}\). (a) Sketch \(\mathbf{u}\) and \(\mathbf{v}\). (b) If \(\mathbf{w}=\mathbf{0}\), show that \(a\) and \(b\) must both be zero. (c) Find \(a\) and \(b\) such that \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}+\mathbf{k}\). (d) Show that no choice of \(a\) and \(b\) yields \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\).

Find the magnitude of \(\mathrm{v}\). Initial point of \(\mathbf{v}:(0,-1,0)\) Terminal point of \(\mathbf{v}:(1,2,-2)\)

In Exercises 91 and 92 , determine the values of \(c\) that satisfy the equation. Let \(\mathbf{u}=\mathbf{i}+\mathbf{2} \mathbf{j}+\mathbf{3} \mathbf{k}\) and \(\mathbf{v}=\mathbf{2} \mathbf{i}+\mathbf{2} \mathbf{j}-\mathbf{k}\) \(\|c \mathbf{v}\|=5\)

Find the distance between the point and the plane.\((0,0,0)\) \(2 x+3 y+z=12\)
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