Convert the point (x, y, z) = (2, — 2, 1) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place. (p, 0, 0) =
Convert the point (x, y, z) = (2, — 2, 1) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place. (p, 0, 0) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Convert the point \((x, y, z) = (2, -2, 1)\) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place.
\((\rho, \theta, \phi) =\) ☐
**Explanation:**
To convert from Cartesian coordinates \((x, y, z)\) to spherical coordinates \((\rho, \theta, \phi)\), use the following formulas:
1. \(\rho = \sqrt{x^2 + y^2 + z^2}\)
2. \(\theta = \arctan{\left(\frac{y}{x}\right)}\)
3. \(\phi = \arccos{\left(\frac{z}{\rho}\right)}\)
Substituting the given values \((x, y, z) = (2, -2, 1)\):
1. Calculate \(\rho\):
\[
\rho = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3
\]
2. Calculate \(\theta\):
\[
\theta = \arctan{\left(\frac{-2}{2}\right)} = \arctan{(-1)} \approx -0.785 \text{ radians (adjust to positive value within range)}
\]
3. Calculate \(\phi\):
\[
\phi = \arccos{\left(\frac{1}{3}\right)} \approx 1.230 \text{ radians}
\]
Adjust \(\theta\) to be positive by adding \(2\pi\) if necessary (e.g., \(\theta \approx 5.498 \text{ radians} \)). Use calculator or trigonometric table for accurate values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21e8a523-579f-40b9-8f0a-805524283a8d%2F10f39741-bd6e-4786-9331-a233a4df6977%2Fc0gw9cb_processed.png&w=3840&q=75)
Transcribed Image Text:Convert the point \((x, y, z) = (2, -2, 1)\) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place.
\((\rho, \theta, \phi) =\) ☐
**Explanation:**
To convert from Cartesian coordinates \((x, y, z)\) to spherical coordinates \((\rho, \theta, \phi)\), use the following formulas:
1. \(\rho = \sqrt{x^2 + y^2 + z^2}\)
2. \(\theta = \arctan{\left(\frac{y}{x}\right)}\)
3. \(\phi = \arccos{\left(\frac{z}{\rho}\right)}\)
Substituting the given values \((x, y, z) = (2, -2, 1)\):
1. Calculate \(\rho\):
\[
\rho = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3
\]
2. Calculate \(\theta\):
\[
\theta = \arctan{\left(\frac{-2}{2}\right)} = \arctan{(-1)} \approx -0.785 \text{ radians (adjust to positive value within range)}
\]
3. Calculate \(\phi\):
\[
\phi = \arccos{\left(\frac{1}{3}\right)} \approx 1.230 \text{ radians}
\]
Adjust \(\theta\) to be positive by adding \(2\pi\) if necessary (e.g., \(\theta \approx 5.498 \text{ radians} \)). Use calculator or trigonometric table for accurate values.
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