What are the spe spherical coordinates of the point whose rectangular Coordinates (1₁ 1₁-5) ? storitreas - 0 = [ P=1
What are the spe spherical coordinates of the point whose rectangular Coordinates (1₁ 1₁-5) ? storitreas - 0 = [ P=1
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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Can anyone please help me with this question please? I am stuck!
![**Title: Conversion of Rectangular to Spherical Coordinates**
**Question:**
What are the spherical coordinates of the point whose rectangular coordinates are \( (4, 1, 5) \)?
**Solution Template:**
For converting from rectangular coordinates \((x, y, z)\) to spherical coordinates \((\rho, \theta, \phi)\), use the following formulas:
1. **\(\rho\) (distance from origin):**
\[
\rho = \sqrt{x^2 + y^2 + z^2}
\]
2. **\(\theta\) (angle in the xy-plane from the x-axis):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
3. **\(\phi\) (angle from the positive z-axis):**
\[
\phi = \cos^{-1}\left(\frac{z}{\rho}\right)
\]
**Explanation of Diagrammatic Representation:**
The image contains placeholders for the spherical coordinates with the symbols \(\phi\), \(\theta\), and \(\rho\) followed by empty boxes, indicating spaces to fill in the calculated values.
**Steps to Rearrange for Audience:**
The viewer is guided through a practical problem-solving approach, requiring them to:
1. Calculate \(\rho\) using the given \(x\), \(y\), and \(z\).
2. Determine \(\theta\) using the arctangent function with \(x\) and \(y\).
3. Compute \(\phi\) using the arccosine function with \(\rho\) and \(z\).
This exercise aims to reinforce understanding of coordinate conversions, beneficial in various fields such as physics and engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F17117971-6394-4852-8185-a5c4df736d94%2Fm5i1c9d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Conversion of Rectangular to Spherical Coordinates**
**Question:**
What are the spherical coordinates of the point whose rectangular coordinates are \( (4, 1, 5) \)?
**Solution Template:**
For converting from rectangular coordinates \((x, y, z)\) to spherical coordinates \((\rho, \theta, \phi)\), use the following formulas:
1. **\(\rho\) (distance from origin):**
\[
\rho = \sqrt{x^2 + y^2 + z^2}
\]
2. **\(\theta\) (angle in the xy-plane from the x-axis):**
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
3. **\(\phi\) (angle from the positive z-axis):**
\[
\phi = \cos^{-1}\left(\frac{z}{\rho}\right)
\]
**Explanation of Diagrammatic Representation:**
The image contains placeholders for the spherical coordinates with the symbols \(\phi\), \(\theta\), and \(\rho\) followed by empty boxes, indicating spaces to fill in the calculated values.
**Steps to Rearrange for Audience:**
The viewer is guided through a practical problem-solving approach, requiring them to:
1. Calculate \(\rho\) using the given \(x\), \(y\), and \(z\).
2. Determine \(\theta\) using the arctangent function with \(x\) and \(y\).
3. Compute \(\phi\) using the arccosine function with \(\rho\) and \(z\).
This exercise aims to reinforce understanding of coordinate conversions, beneficial in various fields such as physics and engineering.
![**Title: Understanding Spherical Coordinates**
**Introduction:**
This educational piece discusses the conversion of rectangular coordinates to spherical coordinates, a common topic in advanced mathematics and physics.
**Problem:**
Determine the spherical coordinates of the point whose rectangular coordinates are (1, 1, -5).
**Solution Steps:**
1. **Representation in Spherical Coordinates:**
- Spherical coordinates \((\rho, \theta, \phi)\) are used to represent points in 3D space, where:
- \(\rho\) is the radial distance from the origin.
- \(\theta\) is the azimuthal angle in the xy-plane from the positive x-axis.
- \(\phi\) is the polar angle from the positive z-axis.
2. **Conversion Formulas:**
- \(\rho = \sqrt{x^2 + y^2 + z^2}\)
- \(\theta = \tan^{-1}(y/x)\)
- \(\phi = \cos^{-1}(z/\rho)\)
3. **Calculations:**
- Given \(x = 1\), \(y = 1\), \(z = -5\):
- Calculate \(\rho\):
\[
\rho = \sqrt{1^2 + 1^2 + (-5)^2} = \sqrt{1 + 1 + 25} = \sqrt{27}
\]
- Calculate \(\theta\):
\[
\theta = \tan^{-1}(1/1) = \tan^{-1}(1) = \frac{\pi}{4}
\]
- Calculate \(\phi\):
\[
\phi = \cos^{-1}(-5/\sqrt{27})
\]
**Conclusion:**
The spherical coordinates \((\rho, \theta, \phi)\) provide an efficient way to describe the location of points in 3D space. In practice, these conversions are essential in fields such as engineering, physics, and computer graphics. Understanding how to convert between these systems enables better comprehension of spatial relationships and vector representation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F17117971-6394-4852-8185-a5c4df736d94%2F03o9hor_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Spherical Coordinates**
**Introduction:**
This educational piece discusses the conversion of rectangular coordinates to spherical coordinates, a common topic in advanced mathematics and physics.
**Problem:**
Determine the spherical coordinates of the point whose rectangular coordinates are (1, 1, -5).
**Solution Steps:**
1. **Representation in Spherical Coordinates:**
- Spherical coordinates \((\rho, \theta, \phi)\) are used to represent points in 3D space, where:
- \(\rho\) is the radial distance from the origin.
- \(\theta\) is the azimuthal angle in the xy-plane from the positive x-axis.
- \(\phi\) is the polar angle from the positive z-axis.
2. **Conversion Formulas:**
- \(\rho = \sqrt{x^2 + y^2 + z^2}\)
- \(\theta = \tan^{-1}(y/x)\)
- \(\phi = \cos^{-1}(z/\rho)\)
3. **Calculations:**
- Given \(x = 1\), \(y = 1\), \(z = -5\):
- Calculate \(\rho\):
\[
\rho = \sqrt{1^2 + 1^2 + (-5)^2} = \sqrt{1 + 1 + 25} = \sqrt{27}
\]
- Calculate \(\theta\):
\[
\theta = \tan^{-1}(1/1) = \tan^{-1}(1) = \frac{\pi}{4}
\]
- Calculate \(\phi\):
\[
\phi = \cos^{-1}(-5/\sqrt{27})
\]
**Conclusion:**
The spherical coordinates \((\rho, \theta, \phi)\) provide an efficient way to describe the location of points in 3D space. In practice, these conversions are essential in fields such as engineering, physics, and computer graphics. Understanding how to convert between these systems enables better comprehension of spatial relationships and vector representation.
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