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Chapter 1: Problem 3

For Exereises \(1-4,\) find the (a) cylindrical and (b) spherical coordinates of the point whose Cartesian coordinates are given. $$ (\sqrt{21},-\sqrt{7}, 0) $$

Short Answer

Expert verified
(a) Cylindrical: \ (2\sqrt{7}, -\pi/6, 0) \ (b) Spherical: \ (2\sqrt{7}, -\pi/6, \pi/2) \

Step by step solution

01

- Convert to cylindrical coordinates

Cylindrical coordinates \(r, \theta, z\) can be obtained from Cartesian coordinates \((x, y, z)\) using the following formulas: \ r = \sqrt{x^2 + y^2} \, \theta = \/atan2(y, x), and \ z = z\.\ Since the given point is \( \sqrt{21}, -\sqrt{7}, 0 \):\ ewline\ - Calculate \ r: \ r = \sqrt{(\sqrt{21})^2 + (-\sqrt{7})^2} = \sqrt{21 + 7} = \sqrt{28} = 2\sqrt{7} \.\ ewline\ - Calculate \ \theta: \ \theta = \/atan2(-\sqrt{7}, \sqrt{21}) = - \/arctan(1/2\sqrt{3}) = -\pi/6 \.\ ewline\ - Calculate z: z remains the same, so \ z = 0 \.
02

- Write the cylindrical coordinates

From Step 1, we found: \r = 2\sqrt{7}, \ \theta = -\pi/6, and z = 0 \. Therefore, cylindrical coordinates are \(2\sqrt{7}, -\pi/6, 0 \).
03

- Convert to spherical coordinates

Spherical coordinates \( \rho, \theta, \phi \) can be obtained from Cartesian coordinates \((x, y, z)\) using the following formulas: \ \rho = \sqrt{x^2 + y^2 + z^2}\text{ , } \theta remains the same as in cylindrical (azimuthal angle), and \phi = \/arccos(z / \rho) \.\ Since the given point is \ ( \sqrt{21}, -\sqrt{7}, 0 ): \ ewline - Calculate \ \rho: \ \rho = \sqrt{ (\sqrt{21})^2 + (-\sqrt{7})^2 + 0^2 } = \sqrt{21 + 7 + 0} = \sqrt{28} = 2\sqrt{7} \. ewline - Calculate \ \phi: \ \phi = \/arccos( 0 / 2\sqrt{7} ) = \/arccos(0) = \pi/2 \.
04

- Write the spherical coordinates

From Step 3, we found: \ \rho = 2\sqrt{7}, \ \theta = -\pi/6, and \ \phi = \pi/2 \. Therefore, spherical coordinates are \(2\sqrt{7}, -\pi/6, \pi/2 \).

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Key Concepts

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

Understanding Cylindrical Coordinates
Cylindrical coordinates extend the 2-dimensional polar coordinates to 3 dimensions by adding a height component.
Instead of \(x, y, z\) as in Cartesian coordinates, we use \(r, \theta, z\) where:
  • \(r\) is the radius or distance from the z-axis
  • \(\theta\) is the azimuthal angle in the xy-plane
  • \(z\) is the same as in Cartesian coordinates
To convert from Cartesian to cylindrical coordinates, we use the following formulas:
\(r = \sqrt{x^2 + y^2}\)
\(\theta = \arctan(y / x)\)
\(z = z\)
Let鈥檚 see how this is applied to the point \(\sqrt{21}, -\sqrt{7}, 0\).
  • Calculate \(r\): \(r = \sqrt{(\sqrt{21})^2 + (-\sqrt{7})^2} = \sqrt{21 + 7} = \sqrt{28} = 2\sqrt{7}\)
  • Determine \(\theta\): \(\theta = \arctan(-\sqrt{7} / \sqrt{21}) = \arctan(-1 / 2\sqrt{3}) = -\pi / 6\)
  • z remains the same: \(z = 0\)
Hence, the cylindrical coordinates for the given point are \(2\sqrt{7}, -\pi/6, 0\).
Introduction to Spherical Coordinates
Spherical coordinates provide a natural way to describe points in three dimensions using \(\rho, \theta, \phi\).
These are particularly useful for problems where symmetry about a point is involved.
  • \(\rho\) is the radial distance from the origin.
  • \(\theta\) is the same azimuthal angle as in cylindrical coordinates.
  • \(\phi\) is the polar angle from the positive z-axis.
Conversion from Cartesian coordinates \(x, y, z\) to spherical coordinates involves:
\(\rho = \sqrt{x^2+y^2+z^2}\),
\(\theta = \arctan(y / x)\),
\(\phi = \arccos(z / \rho)\)
Using the point \(\sqrt{21}, -\sqrt{7}, 0\):
  • Calculate \(\rho: \rho = \sqrt{(\sqrt{21})^2 + (-\sqrt{7})^2 + 0^2} = \sqrt{21 + 7} = \sqrt{28} = 2\sqrt{7}\)
  • Determine \(\phi: \phi = \arccos(0 / 2\sqrt{7}) = \arccos(0) = \pi/2\)
  • Use \(\theta\) from cylindrical coordinates: \(\theta = -\pi/6\)
Therefore, the spherical coordinates are \(2\sqrt{7}, -\pi/6, \pi/2\).
Cartesian Coordinates Explained
Cartesian coordinates (\(x, y, z\)) are the most widely used system for describing points in space.
The system is based on three perpendicular axes: the x-axis, y-axis, and z-axis.
Every point in space is described by its distance from these three axes.
  • The x-coordinate (\(x\)) measures the distance along the x-axis.
  • The y-coordinate (\(y\)) measures the distance along the y-axis.
  • The z-coordinate (\(z\)) measures the distance along the z-axis.
This system is intuitive and simple for many practical applications.
For the given point \(\sqrt{21}, -\sqrt{7}, 0\), each coordinate value directly specifies its position:
  • x is \(\sqrt{21}\)
  • y is \(-\sqrt{7}\)
  • z is \(0\)
To convert this point into other coordinate systems like cylindrical or spherical, we use the relationships and formulas specific to those systems.

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Most popular questions from this chapter

Consider only vectors in \(\mathbb{R}^{3}\) for Exercises \(15-25 .\) Prove or give a counterexample: If \(\mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w},\) then \(\mathbf{v}=\mathbf{w}\)

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