Problem 65 - Advanced Engineering Mathematics

Question

Convert the points given in rectangular coordinates to spherical coordinates. $$ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 1\right) $$

Step-by-Step Solution

Answer

The spherical coordinates are \((\sqrt{2}, \frac{\pi}{6}, \frac{\pi}{4})\).

Step 1: Identify the Rectangular Coordinates

The given rectangular coordinates are \(x = \frac{\sqrt{3}}{2}\), \(y = \frac{1}{2}\), and \(z = 1\).

Step 2: Calculate the Radius \(r\)

In spherical coordinates, \(r\) represents the distance from the origin. It is calculated using the formula \(r = \sqrt{x^2 + y^2 + z^2}\). Thus, we have:\[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + 1^2} = \sqrt{\frac{3}{4} + \frac{1}{4} + 1} = \sqrt{2} \]

Step 3: Calculate the Azimuthal Angle \(\theta\)

\(\theta\) is the angle in the xy-plane from the positive x-axis. It is found using \(\tan \theta = \frac{y}{x}\). Thus:\[ \theta = \tan^{-1}\left(\frac{1/2}{\sqrt{3}/2}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]

Step 4: Calculate the Polar Angle \(\phi\)

\(\phi\) is the angle from the positive z-axis. It is calculated using \(\phi = \cos^{-1}\left(\frac{z}{r}\right)\). With \(z = 1\) and \(r = \sqrt{2}\):\[ \phi = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \]

Step 5: Compile the Spherical Coordinates

The spherical coordinates are given by \((r, \theta, \phi)\). From our earlier calculations, we find:\[ (\sqrt{2}, \frac{\pi}{6}, \frac{\pi}{4}) \]

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of specifying the location of a point in a plane or in space using a numerical combination of axes. In two-dimensional space, these coordinates are designated as \(x, y\), while in three-dimensional space, they are represented as \(x, y, z\).
This system is named after the mathematician René Descartes.
In practice, this coordinate system uses perpendicular axes, usually labeled as the x-axis, y-axis, and z-axis, allowing for easy plotting and understanding of geometric shapes and locations in space.

Key aspects to remember about rectangular coordinates include:

• They provide a clear and intuitive representation of points on a plane or in space.
• They use orthogonal axes, which mean the axes are at right angles to each other.
• Values on these axes are measured as distances from a predefined origin.

Rectangular coordinates are foundational in mathematics and physics, providing a framework that is essential for moving into more complex coordinate systems like spherical coordinates.

Steps in Coordinate Conversion

The process of coordinate conversion from rectangular to spherical coordinates involves translating points from one system to another. This translation gives different perspectives of positioning in space.
It is essential in fields like physics and engineering, where understanding the disposition of a point in three-dimensional space is crucial.

Conversion from rectangular \(x, y, z\) to spherical coordinates \(r, \theta, \phi\) follows these steps:

• Calculate the radius \(r\), the distance from the origin, using \sqrt{x^2 + y^2 + z^2}\. This step establishes the radial distance, offering a debut into spherical space.
• Determine the azimuthal angle \(\theta\), an angle within the xy-plane from the positive x-axis. It's found via \tan \theta = \frac{y}{x}\.
• Measure the polar angle \(\phi\), which is the angle measured from the positive z-axis. Utilize \phi = \cos^{-1}\left(\frac{z}{r}\right)\.

Each of these calculations unveils an aspect of the point's spatial setting in spherical coordinates, permitting a holistic examination of location beyond the rigid rectangular framework.

The Art of Mathematics Problem Solving

Mathematics problem solving shines as an intellectual exercise that nurtures logical thinking and analytical skills. Tackling problems like converting coordinates from one system to another requires a systematic approach and insight into mathematical principles.
A crucial first step in problem solving involves clearly defining the problem by identifying what is known and what needs to be determined.

A helpful strategy includes:

• Comprehending what each given value represents and how it interrelates with other parts of the problem.
• Carefully reviewing each calculation step to ensure correctness and clarity.
• Analyzing the solution in the given context to verify that the derived answer makes logical sense.

By cultivating these skills, students not only solve problems more effectively but also gain a deep understanding of mathematical concepts.
This approach does more than solve a single problem; it builds confidence and equips learners with tools to tackle a wide array of mathematical challenges.