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Chapter 9: Problem 27

In Exercises \(25-28,\) convert the point from spherical coordinates to rectangular coordinates. $$ (5, \pi / 4,3 \pi / 4) $$

Short Answer

Expert verified
The converted rectangular coordinates are \((2.5, 2.5, -3.54)\).

Step by step solution

01

Identify the Spherical Coordinates

First, identify the given spherical coordinates. In this case, they are denoted as \( (r, \theta, \phi) = (5, \pi / 4,3 \pi / 4) \).
02

Apply Spherical to Rectangular Conversion Formula

The general formulas to convert spherical coordinates to rectangular coordinates (denoted as (x, y, z)) are: \[ x= r \sin(\phi) \cos(\theta), \] \[ y= r \sin(\phi) \sin(\theta), \] and \[ z= r\cos(\phi). \] Plug in the given spherical coordinates into each of these formulas.
03

Calculate x Value

For the x-coordinate, substitute \( r = 5\), \( \phi = 3 \pi / 4 \), and \( \theta = \pi / 4 \) into the x-formula: \[ x= 5 \sin(3\pi / 4) \cos(\pi / 4). \] This simplifies to: \[ x= 5(0.7071)(0.7071) = 2.5. \]
04

Calculate y Value

For the y-coordinate, again substitute the given values into the y-formula: \[ y= 5 \sin(3\pi / 4) \sin(\pi / 4). \] This simplifies to the same value as x, so \[ y=2.5. \]
05

Calculate z Value

Finally for z-coordinate, substitute the given values into the z-formula: \[ z= 5\cos(3\pi / 4). \] This simplifies to: \[ z = 5(-0.7071) = -3.54. \]
06

Final Rectangular Coordinates

Combine the calculated x, y, z values to give the final rectangular coordinates.

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Key Concepts

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

Coordinate Conversion
Converting coordinates from one system to another is a fundamental part of multivariable calculus and geometry. In this context, we are discussing the conversion from spherical coordinates to rectangular (or Cartesian) coordinates.

To understand this conversion:
  • Spherical coordinates are given as \( (r, \theta, \phi) \), where \((r)\) is the radial distance from the origin, \(\theta\) is the azimuthal angle in the xy-plane, and \(\phi\) is the polar angle from the positive z-axis.
  • Rectangular coordinates are expressed as \( (x, y, z) \), representing the location along the standard x, y, and z-axes in 3D space.
This conversion involves transforming the angles and distance from spherical terms into the straightforward x, y, and z format of rectangular coordinates. This is often critical in fields like physics and engineering.
Spherical Coordinates
Spherical coordinates provide a way to describe the position of a point in three-dimensional space with three parameters.

The parameters in spherical coordinates include:
  • Radial distance (\(r\)): This measures how far a point is from the origin. In our example, \(r = 5\).
  • Azimuthal angle (\(\theta\)): This angle, in the xy-plane from the positive x-axis, is given as \(\pi/4\) radians, indicating the point's directional orientation.
  • Polar angle (\(\phi\)): Known also as the inclination angle, this is \(\phi = 3\pi/4 \), measuring the angle down from the positive z-axis.
These coordinates are especially useful in scenarios where systems have a natural radial symmetry, such as planets, stars, and atoms.
Rectangular Coordinates
Rectangular coordinates are more familiar to most people as they describe points in space using the standard x, y, and z axes.

To convert from spherical to rectangular coordinates, we use the following formulas:
  • x-coordinate: \(x = r \sin(\phi) \cos(\theta)\)
  • y-coordinate: \(y = r \sin(\phi) \sin(\theta)\)
  • z-coordinate: \(z = r \cos(\phi)\)
By substituting the spherical coordinates \( (5, \pi/4, 3\pi/4) \), we calculated:
  • \(x = 2.5\)
  • \(y = 2.5\)
  • \(z = -3.54\)
This transformation is vital in various scientific applications, simplifying complex spherical systems into manageable rectangular forms.

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