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Chapter 11: Problem 11

Convert from spherical to cylindrical coordinates. (a) \((5, \pi / 4,2 \pi / 3)\) (b) \((1,7 \pi / 6, \pi)\) (c) \((3,0,0)\) (d) \((4, \pi / 6, \pi / 2)\)

Short Answer

Expert verified
(a) \((5\sqrt{3}/2, \pi/4, -5/2)\), (b) \((0, 7\pi/6, -1)\), (c) \((0, 0, 3)\), (d) \((4, \pi/6, 0)\)."

Step by step solution

01

Understand the Coordinate Conversion Formulas

To convert from spherical to cylindrical coordinates, we use the following formulas: - For cylindrical coordinates \(r, \theta, z\), the corresponding spherical coordinates \(\rho, \phi, \theta\) are related as follows: 1. \(r = \rho \sin \phi\)2. \(\theta = \theta\) (same in both systems)3. \(z = \rho \cos \phi\)
02

Convert (a) \\( (5, \pi / 4, 2\pi / 3) \\\)

First, identify the spherical coordinates \(\rho = 5, \phi = 2\pi / 3, \theta = \pi / 4\).1. Compute \(r = \rho \sin \phi = 5 \sin(2\pi / 3) = 5(\sqrt{3}/2) = 5\sqrt{3}/2\).2. \(\theta = \pi / 4\) remains the same.3. Compute \(z = \rho \cos \phi = 5 \cos(2\pi / 3) = 5(-1/2) = -5/2\).Hence, the cylindrical coordinates are \((5\sqrt{3}/2, \pi / 4, -5/2)\).
03

Convert (b) \\( (1, 7\pi / 6, \pi) \\\)

First, identify the spherical coordinates \(\rho = 1, \phi = \pi, \theta = 7\pi / 6\).1. Compute \(r = \rho \sin \phi = 1 \sin(\pi) = 0\).2. \(\theta = 7\pi / 6\) remains the same.3. Compute \(z = \rho \cos \phi = 1 \cos(\pi) = -1\).Hence, the cylindrical coordinates are \( (0, 7\pi/6, -1) \).
04

Convert (c) \\( (3, 0, 0) \\\)

First, identify the spherical coordinates \(\rho = 3, \phi = 0, \theta = 0\).1. Compute \(r = \rho \sin \phi = 3 \sin(0) = 0\).2. \(\theta = 0\) remains the same.3. Compute \(z = \rho \cos \phi = 3 \cos(0) = 3\).Hence, the cylindrical coordinates are \((0, 0, 3)\).
05

Convert (d) \\( (4, \pi / 6, \pi / 2) \\\)

First, identify the spherical coordinates \(\rho = 4, \phi = \pi/2, \theta = \pi/6\).1. Compute \(r = \rho \sin \phi = 4 \sin(\pi/2) = 4(1) = 4\).2. \(\theta = \pi/6\) remains the same.3. Compute \(z = \rho \cos \phi = 4 \cos(\pi/2) = 4(0) = 0\).Hence, the cylindrical coordinates are \( (4, \pi/6, 0) \).

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Key Concepts

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

Spherical Coordinates
Spherical coordinates are a way to represent points in three-dimensional space using three values: \( (\rho, \phi, \theta) \). Each of these components has a specific geometric meaning, making this system particularly useful in fields like physics and engineering.

Here is what each term represents:
  • \( \rho \) (rho) is the radial distance from the origin to the point.
  • \( \phi \) (phi) is the polar angle, measured from the positive z-axis downward.
  • \( \theta \) (theta) is the azimuthal angle in the xy-plane, measured from the positive x-axis.
Understanding how these components interact is essential for converting between coordinate systems. To visualize or convert spherical coordinates, it's often helpful to picture the angles wrapping around or descending from the origin outward, like the lines of latitude and longitude on a globe.
Cylindrical Coordinates
Cylindrical coordinates provide another way to describe positions in three-dimensional space using \((r, \theta, z)\), blending elements of polar and Cartesian coordinates.

Here's how each parameter is defined:
  • \( r \) is the radial distance from the z-axis to the point's projection in the xy-plane.
  • \( \theta \) (theta) is the angle that the radial line makes with the positive x-axis, similar to the angle in polar coordinates.
  • \( z \) represents the height above or below the xy-plane, just like in Cartesian coordinates.
Converting from spherical to cylindrical coordinates involves specific formulas:
  • Compute \( r \) using \( r = \rho \sin \phi \).
  • Maintain \( \theta \) as it remains the same.
  • Calculate \( z \) with \( z = \rho \cos \phi \).
By understanding these relationships, it becomes easier to perform conversions accurately and see how each set of coordinates relates spatially.
Trigonometric Functions
Trigonometric functions are mathematical functions that relate angles to side ratios in right-angled triangles. They are crucial in converting coordinates because they offer a natural way to deal with angles and periodic patterns.

The two main functions used in coordinate conversions are:
  • Sine \( \sin \phi \): Determines the component of the radius that lies in the xy-plane. It's used in finding the radial distance \( r \) when converting from spherical to cylindrical coordinates.
  • Cosine \( \cos \phi \): Provides the height component by dictating how much of the radius aligns with the z-axis. In conversions, it determines \( z \).
Understanding these functions' symmetries and periodic behavior (e.g., how sine and cosine values oscillate between -1 and 1) helps predict how points translate between different systems. Their graphical representations as wave-like patterns further underscore their utility in a wide range of mathematical and physical applications.

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Most popular questions from this chapter

An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ z=r^{2} $$

Find the distance between the given parallel planes. $$ \begin{aligned} &x+y+z=1 \\ &x+y+z=-1 \end{aligned} $$

Show that if \(a, b\), and \(c\) are nonzero, then the plane whose intercepts with the coordinate axes are \(x=a\), \(y=b\), and \(z=c\) is given by the equation $$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$

Find the distance between the given parallel planes. $$ \begin{array}{r} -2 x+y+z=0 \\ 6 x-3 y-3 z-5=0 \end{array} $$

If \(L\) is a line in 2 -space or 3 -space that passes through the points \(A\) and \(B\), then the distance from a point \(P\) to the line \(L\) is equal to the length of the component of the vector \(\overrightarrow{A P}\) that is orthogonal to the vector \(\overrightarrow{A B}\) (see the accompanying figure). Use this result to find the distance from the point \(P(1,0)\) to the line through \(A(2,-3)\) and \(B(5,1)\).
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