Question

Points are given in either the rectangular, cylindrical or spherical coordinate systems. Find the coordinates of the points in the other systems. (a) Points in rectangular coordinates: (2,2,1) and \((-\sqrt{3}, 1,0)\) (b) Points in cylindrical coordinates: \((2, \pi / 4,2)\) and \((3,3 \pi / 2,-4)\) (c) Points in spherical coordinates: \((2, \pi / 4, \pi / 4)\) and (1,0,0)

Short answer

All coordinates converted for each point: first set to cylindrical/spherical, second set to rectangular/spherical, and third set to rectangular/cylindrical.

Step-by-step solution

Rectangular to Cylindrical/Spherical (Point 1)

For the point (2,2,1), convert to cylindrical coordinates using the formulas:- Radial distance: \(r = \sqrt{x^2 + y^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\)- Angle: \(\theta = \tan^{-1}(\frac{y}{x}) = \tan^{-1}(\frac{2}{2}) = \frac{\pi}{4}\)- Height (z remains unchanged): \(z = 1\)Cylindrical coordinates: \((2\sqrt{2}, \frac{\pi}{4}, 1)\)Convert to spherical coordinates:- Radius: \(\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{2^2 + 2^2 + 1^2} = \sqrt{9} = 3\)- Polar angle: \(\phi = \cos^{-1}(\frac{z}{\rho}) = \cos^{-1}(\frac{1}{3})\)- Azimuthal angle: \(\theta = \tan^{-1}(\frac{y}{x}) = \frac{\pi}{4}\)Spherical coordinates: \((3, \frac{\pi}{4}, \cos^{-1}(\frac{1}{3}))\)

Rectangular to Cylindrical/Spherical (Point 2)

For the point \((-\sqrt{3}, 1, 0)\), convert to cylindrical coordinates:- Radial distance: \(r = \sqrt{x^2 + y^2} = \sqrt{(-\sqrt{3})^2 + 1^2} = 2\)- Angle: \(\theta = \tan^{-1}(\frac{y}{x}) = \tan^{-1}(\frac{1}{-sqrt{3}}) = -\frac{pi}{6}\) (in the correct quadrant: \(\theta = \frac{5\pi}{6}\))- Height: \(z = 0\)Cylindrical coordinates: \((2, \frac{5\pi}{6}, 0)\)Convert to spherical coordinates:- Radius: \(\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{3 + 1} = 2\)- Polar angle: \(\phi = \cos^{-1}(\frac{z}{\rho}) = \cos^{-1}(0) = \frac{\pi}{2}\)- Azimuthal angle: \(\theta = \frac{5\pi}{6}\)Spherical coordinates: \((2, \frac{5\pi}{6}, \frac{\pi}{2})\)

Cylindrical to Rectangular/Spherical (Point 1)

For the point \((2, \pi/4, 2)\) in cylindrical coordinates, convert to rectangular coordinates:- \(x = r\cos\theta = 2\cos(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}\)- \(y = r\sin\theta = 2\sin(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}\)- \(z = 2\)Rectangular coordinates: \((\sqrt{2}, \sqrt{2}, 2)\)Convert to spherical coordinates:- \(\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\)- \(\theta = \pi/4\)- \(\phi = \cos^{-1}(\frac{z}{\rho}) = \cos^{-1}(\frac{2}{2\sqrt{2}}) = \frac{\pi}{4}\)Spherical coordinates: \((2\sqrt{2}, \pi/4, \pi/4)\)

Cylindrical to Rectangular/Spherical (Point 2)

For the point \((3, 3\pi/2, -4)\) in cylindrical coordinates, convert to rectangular coordinates:- \(x = r\cos\theta = 3\cos(3\pi/2) = 3 \cdot 0 = 0\)- \(y = r\sin\theta = 3\sin(3\pi/2) = 3 \cdot (-1) = -3\)- \(z = -4\)Rectangular coordinates: \((0, -3, -4)\)Convert to spherical coordinates:- \(\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{0^2 + (-3)^2 + (-4)^2} = 5\)- \(\theta = 3\pi/2\)- \(\phi = \cos^{-1}(\frac{z}{\rho}) = \cos^{-1}(\frac{-4}{5})\)Spherical coordinates: \((5, 3\pi/2, \cos^{-1}(-4/5))\)

Spherical to Rectangular/Cylindrical (Point 1)

For the point \((2, \pi/4, \pi/4)\) in spherical coordinates, convert to rectangular coordinates:- \(x = \rho\sin\phi\cos\theta = 2\sin(\pi/4)\cos(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} = 1\)- \(y = \rho\sin\phi\sin\theta = 2\sin(\pi/4)\sin(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} = 1\)- \(z = \rho\cos\phi = 2\cos(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}\)Rectangular coordinates: \((1, 1, \sqrt{2})\)Convert to cylindrical coordinates:- \(r = \sqrt{x^2 + y^2} = \sqrt{1^2 + 1^2} = \sqrt{2}\)- \(\theta = \pi/4\)- \(z = \sqrt{2}\)Cylindrical coordinates: \((\sqrt{2}, \pi/4, \sqrt{2})\)

Spherical to Rectangular/Cylindrical (Point 2)

For the point \((1, 0, 0)\) in spherical coordinates, convert to rectangular coordinates:- \(x = \rho\sin\phi\cos\theta = 1\cdot0\cdot1 = 0\)- \(y = \rho\sin\phi\sin\theta = 1\cdot0\cdot0 = 0\)- \(z = \rho\cos\phi = 1\cdot1 = 1\)Rectangular coordinates: \((0, 0, 1)\)Convert to cylindrical coordinates:- \(r = \sqrt{x^2 + y^2} = 0\)- \(\theta\) can be any value because \( r = 0 \), conventionally 0- \(z = 1\)Cylindrical coordinates: \((0, 0, 1)\)

Key concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common type of coordinate system. They allow us to specify the position of a point in three-dimensional space using three values: \(x\), \(y\), and \(z\). This system is particularly intuitive because it uses perpendicular axes to represent dimensions. For example:

• \(x\) represents the horizontal distance along the x-axis.
• \(y\) represents the horizontal distance along the y-axis.
• \(z\) represents the vertical distance from the xy-plane.

When we're given a point in rectangular coordinates, such as \(2, 2, 1\), it tells us precisely how far the point is from each of the three axes.
This system is widely used because it aligns so well with our natural understanding of up, down, left, right, and depth.

Cylindrical Coordinates

Cylindrical coordinates provide a way to specify points in three-dimensional space by combining the polar coordinate system and a height value. They are particularly useful for problems with rotational symmetry around a central axis. Cylindrical coordinates are denoted by \( (r, \theta, z) \), where:

• \(r\) is the radial distance from the z-axis.
• \(\theta\) is the angle between the projection of the point in the xy-plane and the positive x-axis.
• \(z\) is the height above the xy-plane, which is the same as in rectangular coordinates.

To convert from rectangular coordinates to cylindrical coordinates, the following formulas are used:
\[ r = \sqrt{x^2 + y^2} \]
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
This system efficiently tackles 3D problems like determining the position of a point on a cylinder, where the height and radial distance are easily defined.

Spherical Coordinates

The spherical coordinate system allows us to describe a point in 3D space with three values based on total distance and angles from a reference point. Usual notation for spherical coordinates is \( (\rho, \theta, \phi) \), where:

• \(\rho\) is the radial or spherical distance from the origin to the point.
• \(\theta\) is the azimuthal angle, taken from the positive x-axis within the xy-plane.
• \(\phi\) is the polar angle or inclination measured from the positive z-axis.

Converting from rectangular to spherical coordinates uses these formulas:
\[ \rho = \sqrt{x^2 + y^2 + z^2} \]
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
\[ \phi = \cos^{-1}\left(\frac{z}{\rho}\right) \]
This system is beneficial for applications involving radii and directions from a central point, like celestial coordinates in astronomy.