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Chapter 2: Problem 410

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. $$\left(9,-\frac{\pi}{6}, \frac{\pi}{3}\right)$$

Short Answer

Expert verified
Cylindrical coordinates: \(\left(\frac{9\sqrt{3}}{2}, -\frac{\pi}{6}, \frac{9}{2}\right)\).

Step by step solution

01

Understand Spherical Coordinates

Spherical coordinates are given as \(( ho, heta, ext{and } eta)\).In this problem, \(\rho = 9\) is the radial distance from the origin, \(\theta = -\frac{\pi}{6}\) is the azimuthal angle in the \(xy\)-plane from the positive \(x\)-axis, and \(\beta = \frac{\pi}{3}\) is the polar angle from the positive \(z\)-axis.
02

Convert Spherical to Cylindrical

To convert from spherical to cylindrical coordinates:- The radial distance in the \(xy\)-plane \( r = \rho \sin(\beta) = 9\sin\left(\frac{\pi}{3}\right) = 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2} \).- The azimuthal angle \(\theta = -\frac{\pi}{6}\) remains the same.- Calculate the height \( z = \rho \cos(\beta) = 9\cos\left(\frac{\pi}{3}\right) = 9 \times \frac{1}{2} = \frac{9}{2} \).Thus, the cylindrical coordinates are \(\left(\frac{9\sqrt{3}}{2}, -\frac{\pi}{6}, \frac{9}{2}\right)\).

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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: \( \rho \), \( \theta \), and \( \beta \). These help to specify locations like your usual Cartesian system but in a more intuitive manner for spherical shapes.
  • \( \rho \): This is the radial distance from the origin to the point. It tells you how far out into space the point is.
  • \( \theta \): Known as the azimuthal angle, it represents the angle in the \( xy \)-plane from the positive \( x \)-axis.
  • \( \beta \): The polar angle from the positive \( z \)-axis, it shows how far the point is tilted from the vertical.

Understanding these angles and distances is crucial, as they define where the point is located in a spherical fashion, much like how we talk about latitude and longitude on a globe.
Conversion Between Coordinate Systems
Moving from one coordinate system to another can seem tricky, but it is quite manageable with a few formulae. When you're converting spherical coordinates \((\rho, \theta, \beta)\) to cylindrical coordinates \((r, \theta, z)\), here's the approach:
  • The radial distance \( r \): This is found by multiplying \( \rho \) by the sine of the polar angle \( \beta \). Mathematically, \( r = \rho \sin(\beta) \).
  • The azimuthal angle \( \theta \): This remains the same in both systems.
  • The height \( z \): This is calculated by multiplying \( \rho \) with the cosine of \( \beta \), yielding \( z = \rho \cos(\beta) \).

These conversions maintain the point's position but express it differently, emphasizing understanding how different system parameters relate. Practice over time can help you smoothly switch perspectives when required.
Radial Distance in Cylindrical Coordinates
The radial distance in cylindrical coordinates, represented as \( r \), describes how far outward a point extends from the \( z \)-axis.This notion is quite similar to the radial arm of a compass in the police dramas you might have seen. In cylindrical coordinates, this isn't related to the origin directly.
  • Unlike in spherical coordinates, here \( r \) is a projection onto the \( xy \)-plane.
  • The calculation uses the sine of the polar angle (from spherical coordinates) to find how far out, in a flat line, the point lies from directly above or below it on the \( xy \)-plane.

Calculating \( r \) provides a key piece of the puzzle in positioning, serving as the cylinder's radius from its axis.
Polar Angle in Spherical Coordinates
The polar angle \( \beta \) in spherical coordinates has a unique role. It defines how far the point deviates from the vertical axis, which is often the \( z \)-axis in three dimensions.
  • Measured from the positive \( z \)-axis, it determines the tilting degree downward toward the \( xy \)-plane.
  • A smaller \( \beta \) means the point is closer to the top鈥攏ear the "ceiling" of the sphere.
  • As \( \beta \) increases, the point tilts more toward the "equator" or middle of the sphere.

Think of the polar angle as controlling the lean of a skyscraper 鈥� greater angles mean bending over more toward the ground. Comprehending \( \beta \) is crucial when working between spherical and cylindrical representations.

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Most popular questions from this chapter

For the following exercises, point \(P\) and vector \(\mathbf{v}\) are given. Let \(L\) be the passing through point \(P\) with direction \(\mathbf{v} .\) a. Find parametric equations of line \(L .\) b. Find symmetric equations of line \(L\) . c. Find the intersection of the line with the \(x y\) -plane. \(P(3,1,5), \mathbf{v}=\overrightarrow{Q R}\) Where \(Q(2,2,3)\) and \(R(3,2,3)\)

The following problems consider your unsuccessful attempt to take the tire off your car using a wrench to loosen the bolts. Assume the wrench is 0.3 \(\mathrm{m}\) long and you are able to apply a 200 -N force. Someone lends you a tire jack and you are now able to apply a 200 -N force at an \(80^{\circ}\) angle. Is your resulting torque going to be more or less? What is the new resulting torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

In the following exercises, points \(P\) and \(Q\) are given. Let \(L\) be the line passing through points \(P\) and \(Q .\) a. Find the vector equation of line \(L\) b. Find parametric equations of line \(L\) c. Find symmetric equations of line \(L\) d. Find parametric equations of the line segment determined by \(P\) and \(Q\) . \(P(7,-2,6), \quad Q(-3,0,6)\)

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. \(L_{1} : x=2, y=1, z=t\) \(L_{2} : x=1, y=1, z=2-3 t, \quad t \in \mathbb{R}\)

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(3 x+4 y-12=0\)
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