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Chapter 2: Problem 364

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates \((r, \theta, z)\) of a point are given. Find the rectangular coordinates \((x, y, z)\) of the point. \(\left(3, \frac{\pi}{3}, 5\right)\)

Short Answer

Expert verified
The rectangular coordinates are \((1.5, \frac{3\sqrt{3}}{2}, 5)\).

Step by step solution

01

Understand Cylindrical to Rectangular Conversion

To convert from cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), we use the formulas: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
02

Apply Formula for x

Using the cylindrical coordinates \((3, \frac{\pi}{3}, 5)\), we calculate \(x\) by substituting the values into the formula:\[x = 3 \cos\left(\frac{\pi}{3}\right) = 3 \times \frac{1}{2} = 1.5\]
03

Apply Formula for y

Now, calculate \(y\) using the formula and the given coordinates:\[y = 3 \sin\left(\frac{\pi}{3}\right) = 3 \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}\]
04

Assign z Value

In cylindrical coordinates, the \(z\)-value remains unchanged when converting to rectangular coordinates. Therefore, \(z = 5\).
05

Compile the Rectangular Coordinates

Based on the calculations, the rectangular coordinates are:\((x, y, z) = \left(1.5, \frac{3\sqrt{3}}{2}, 5\right)\)

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Key Concepts

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

Cylindrical Coordinates
Cylindrical coordinates are a three-dimensional coordinate system often used in situations where symmetry about an axis, typically the z-axis, is present. The system extends the two-dimensional polar coordinate system by adding a height component. Instead of using the traditional Cartesian x, y, and z to describe a point in space, cylindrical coordinates use:
  • \(r\): the radial distance from the origin to the point in the xy-plane
  • \(\theta\): the angle in the xy-plane measured from the positive x-axis, counterclockwise
  • \(z\): the height above the xy-plane
For example, given the cylindrical coordinates \((3, \frac{\pi}{3}, 5)\),
  • \(r = 3\)
  • \(\theta = \frac{\pi}{3}\) (which is 60 degrees)
  • \(z = 5\)
These values describe a point located three units away from the origin, forming a 60-degree angle with the positive x-axis, and five units above the xy-plane.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are the most common way to describe a point in space. This coordinate system uses a grid of right angles created by three mutually perpendicular axes: x, y, and z.
  • x-coordinate: Distance along the horizontal axis
  • y-coordinate: Distance along the vertical axis
  • z-coordinate: Altitude or height above/below the xy-plane
Each point in three-dimensional space is described uniquely by its x, y, and z values. In our specific conversion problem, we used these formulas:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
  • \(z\) remains unchanged from cylindrical to rectangular
These simple mathematical equations allow students to map from cylindrical to rectangular coordinates, offering a practical approach for many scientific and engineering applications.
Coordinate Conversion
Coordinate conversion is a key mathematical tool that bridges different ways of visualizing and understanding geometric locations. It allows us to switch from one coordinate system to another, simplifying calculations depending on the problem's context. In cases where we switch from cylindrical to rectangular coordinates:First, decode the radial distance and angular position using:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
Using our given coordinates \((3, \frac{\pi}{3}, 5)\), we apply these conversions:
  • For x, \( 3 \cos(\frac{\pi}{3}) = 1.5 \)
  • For y, \( 3 \sin(\frac{\pi}{3}) = \frac{3\sqrt{3}}{2} \)
Finally, \(z\) remains unchanged. This straightforward transformation simplifies complex problems, making them easier to tackle by choosing a more convenient framework for computation and visualization.

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Most popular questions from this chapter

For the following exercises, find the area or volume of the given shapes. The \(\quad\) parallelepiped \(\quad\) formed \(\quad\) by \(\mathbf{a}=\langle 1,4,1\rangle\) and \(\mathbf{b}=\langle 3,6,2\rangle,\) and \(\mathbf{c}=\langle-2,1,-5\rangle\)

For the following exercises, the equation of a quadric surface is given. a. Use the method of completing the square to write the equation in standard form. b. Identify the surface. \(x^{2}+\frac{y^{2}}{4}-\frac{z^{2}}{3}+6 x+9=0\)

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle \(\varphi\) in radians rounded to four decimal places. [T] \((5, \pi, 12)\)

[T] A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the \(z\) -axis as its axis of symmetry is given by \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1, \quad\) where \(a\) and \(c\) are positive real numbers. The spheroid is called oblate if \(ca\). a. The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where \(a=8.7 \mathrm{~mm}\) and \(c=9.6 \mathrm{~mm} .\) Write the equation of the spheroid that models the cornea and sketch the surface. b. Give two examples of objects with prolate spheroid shapes.

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. For vectors \(\mathbf{a}\) and \(\mathbf{b}\) and any given scalar \(c\), \(c(\mathbf{a} \times \mathbf{b})=(c \mathbf{a}) \times \mathbf{b}\)
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