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Chapter 1: Problem 42

For the following exercises, the rectangular coordinates \((x, y, z)\) of a point are given. Find the spherical coordinates \((\rho, \theta, \varphi)\) of the point. Express the measure of the angles in degrees rounded to the nearest integer.\((-1,2,1)\)

Short Answer

Expert verified
The spherical coordinates are \( (\sqrt{6}, 117掳, 66掳) \).

Step by step solution

01

Calculate 蟻

The spherical coordinate \(\rho\) is the distance from the origin to the point \((x, y, z)\). It is given by the formula \(\rho = \sqrt{x^2 + y^2 + z^2}\).Substitute \(x = -1, y = 2, \text{ and } z = 1\):\[\rho = \sqrt{(-1)^2 + (2)^2 + (1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}\]
02

Calculate 胃

The spherical coordinate \(\theta\) is the angle in the \(xy\)-plane, measured from the positive \(x\)-axis. It is given by \(\theta = \arctan\left(\frac{y}{x}\right)\).Substitute \(x = -1\) and \(y = 2\):\[\theta = \arctan\left(\frac{2}{-1}\right) = \arctan(-2)\]Since \(\theta\) is in the second quadrant, adjust by adding 180掳:\[\theta = \arctan(-2) + 180掳 \approx 116.57掳\]Rounding to the nearest integer, we get \(\theta \approx 117掳\).
03

Calculate 蠁

The spherical coordinate \(\varphi\) is the angle from the positive \(z\)-axis. It is given by \(\varphi = \arccos\left(\frac{z}{\rho}\right)\).Substitute \(z = 1\) and \(\rho = \sqrt{6}\):\[\varphi = \arccos\left(\frac{1}{\sqrt{6}}\right)\]Calculate \(\varphi\):\[\varphi \approx 65.90掳\]Rounding to the nearest integer, \(\varphi \approx 66掳\).

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are used to describe a point's position in a three-dimensional space using three values:
  • Usually, these values are denoted as
    • \(x\), representing the horizontal position.
    • \(y\), indicating the position along the vertical plane.
    • \(z\), specifying the height of the point above the reference plane.
  • These coordinates are often used because they directly correspond to the measurements you'd take in a 3D environment, much like a graph on a piece of paper lifted into space.
In the exercise, the given point is
  • \((-1, 2, 1)\), meaning it lies one unit to the left of the origin, two units forward, and one unit above the reference xy-plane.
Understanding these coordinates is essential before moving on to spherical coordinates, as they form the starting point for transforming into other coordinate systems.
Coordinate Transformation
Coordinate transformation involves converting one type of coordinates into another, allowing us to represent the same point in different ways. Transforming from rectangular to spherical coordinates provides a different insight into the spatial relationship:
  • The transformation involves three main components:
    • Distance from the origin
      • \(\rho\), calculated using the formula \(\rho = \sqrt{x^2 + y^2 + z^2}\).
    • Angle from the x-axis in the xy-plane
      • \(\theta\), found through \(\theta = \arctan\left(\frac{y}{x}\right)\).
    • Angle from the z-axis
      • \(\varphi\), computed as \(\varphi = \arccos\left(\frac{z}{\rho}\right)\).
  • By using these transformations, we can visualize how the points align with spherical layers and planes in a three-dimensional space, akin to pinpointing a place on a globe using latitude, longitude, and radius.
This exercise guides you step-by-step through each computation, helping you master the conversion process.
Angle Measurement
In spherical coordinates, angle measurement is crucial for understanding the position relative to the axes:
  • \(\theta\) represents the angle in the xy-plane, measured counterclockwise from the positive x-axis.
    • It has to account for direction and position of the point within the coordinate plane. For example, in the second quadrant, an adjustment of adding 180掳 is needed.
  • \(\varphi\) is the angle of elevation from the positive z-axis toward the point.
    • This involves calculating the arc cosine of the ratio between the z-coordinate and the radial distance \(\rho\).
  • These angles are often converted to degrees, making the representation of spherical coordinates intuitive and easily understandable.
It's important to round the angles to convenient units, like the exercise does to the nearest integer, ensuring precise communication in practical applications.

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Most popular questions from this chapter

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