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Magazine Chapter 10: Problem 6
Find the spherical polar coordinates \((r, \theta, \phi)\) of the points: $$ (x, y, z)=(1,2,2) $$
Short Answer
Expert verified
The spherical coordinates are \((3, \arccos(\frac{2}{3}), \arctan(2))\).
Step by step solution
01
Understanding Spherical Coordinates
In spherical coordinates, a point in 3D space is represented by three values: \(r\), the radial distance from the origin; \(\theta\), the polar angle measured from the positive \(z\)-axis; and \(\phi\), the azimuthal angle in the \(xy\)-plane from the positive \(x\)-axis.
02
Calculating the Radial Distance \(r\)
The radial distance \(r\) is calculated using the formula \(r = \sqrt{x^2 + y^2 + z^2}\). For the point \((x, y, z) = (1, 2, 2)\), calculate:\[r = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3.\]
03
Calculating the Polar Angle \(\theta\)
The polar angle \(\theta\) is the angle from the positive \(z\)-axis and is computed using \(\theta = \arccos\left(\frac{z}{r}\right)\). For our point, this is:\[\theta = \arccos\left(\frac{2}{3}\right).\]
04
Calculating the Azimuthal Angle \(\phi\)
The azimuthal angle \(\phi\) is measured in the \(xy\)-plane from the \(x\)-axis given by \(\tan\phi = \frac{y}{x}\). Therefore:\[\phi = \arctan\left(\frac{2}{1}\right) = \arctan(2).\]
05
Assembling the Spherical Coordinates
Now that we have calculated \(r\), \(\theta\), and \(\phi\), we can write the spherical coordinates as:\[(r, \theta, \phi) = \left(3, \arccos\left(\frac{2}{3}\right), \arctan(2)\right).\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radial Distance
In spherical polar coordinates, the radial distance, denoted as \( r \), is essential in determining the position of a point in 3D space. Radial distance is the straight line distance from the origin to the point in question. Imagine it as the length of a line drawn directly from the central point, where the three axes intersect, to the position of interest.
- To calculate \( r \), you employ the formula: \[ r = \sqrt{x^2 + y^2 + z^2} \] This tells you how far the point is from the origin in a 3D space.
- For example, for a point \((x, y, z) = (1, 2, 2)\), the radial distance is computed as:\( r = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3 \).
Polar Angle
The polar angle, represented as \( \theta \), is a key component of spherical coordinates. It measures the angle a point makes with the positive \( z \)-axis. Think of it as the tilt of the point from the top, or how far it rotates downward from the \( z \)-axis, like the tilt of Earth relative to its poles.
- To find \( \theta \), you use the relationship: \[ \theta = \arccos\left(\frac{z}{r}\right) \]
- For the point \( (1, 2, 2) \), with \( r = 3 \), it becomes: \( \theta = \arccos\left(\frac{2}{3}\right) \).
Azimuthal Angle
The azimuthal angle, symbolized as \( \phi \), describes the rotation around the vertical axis within the \( xy \)-plane. Picture moving around a circle on a flat floor; this angle, \( \phi \), indicates how far and in what direction point has rotated starting from the positive \( x \)-axis.
- To compute \( \phi \), the formula: \[ \phi = \arctan\left(\frac{y}{x}\right) \] is used to measure the circular path within the \( xy \)-plane.
- For our specific example, \( (1, 2, 2) \), it simplifies to: \( \phi = \arctan(2) \).
3D Space Representation
Spherical polar coordinates provide a powerful way to represent points in 3D space. They consist of three values: the radial distance \( r \), the polar angle \( \theta \), and the azimuthal angle \( \phi \). This coordinate system is especially useful when dealing with physical problems where symmetry around a point (like the center of a sphere) is relevant.
- The radial distance \( r \) gives you the magnitude or extent of the line from the origin to the point.
- The polar angle \( \theta \) informs how far the point diverges from the vertical axis.
- The azimuthal angle \( \phi \) indicates the direction of the point along the horizontal plane.
