Problem 66 - Advanced Engineering Mathematics

Question

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ \left(-\frac{\sqrt{3}}{2}, 0,-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The spherical coordinates are \((1, \frac{2\pi}{3}, 0)\).

Step 1: Identify Rectangular Coordinates

The given rectangular coordinates are \( x = -\frac{\sqrt{3}}{2} \), \( y = 0 \), and \( z = -\frac{1}{2} \).

Step 2: Calculate the Radius \( r \)

To find the radius \( r \), use the formula \( r = \sqrt{x^2 + y^2 + z^2} \). Plugging in the values, we have:\[ r = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + 0^2 + \left(-\frac{1}{2}\right)^2} \]\[ r = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \].

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

The inclination angle \( \theta \) is calculated using \( \theta = \arccos\left( \frac{z}{r} \right) \). Since \( r = 1 \) and \( z = -\frac{1}{2} \):\[ \theta = \arccos\left( -\frac{1}{2} \right) \]\( \theta = \frac{2\pi}{3} \) radians.

Step 4: Calculate the Azimuthal Angle \( \varphi \)

The azimuthal angle \( \varphi \) is calculated using \( \varphi = \arctan\left( \frac{y}{x} \right) \). Since \( x = -\frac{\sqrt{3}}{2} \) and \( y = 0 \):\[ \varphi = \arctan\left( \frac{0}{-\frac{\sqrt{3}}{2}} \right) \]Thus, \( \varphi = 0 \) radians.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are used to represent points in a 3-dimensional space. This system uses three values:

• The x-coordinate
• The y-coordinate
• The z-coordinate

These coordinates are used to describe the location of a point relative to a set of orthogonal (perpendicular) axes. For example, in the problem, the given rectangular coordinates are \( x = -\frac{\sqrt{3}}{2} \), \( y = 0 \), and \( z = -\frac{1}{2} \).
Rectangular coordinates are convenient for solving many mathematical problems, but sometimes, it's beneficial to convert them to other systems like spherical coordinates, especially for problems involving symmetry about a point or a sphere.

Radius Calculation

The radius in spherical coordinates, represented by \( r \), describes the distance from the origin to the point in question. To find this radius, we rely on the formula used in the solution: \( r = \sqrt{x^2 + y^2 + z^2} \).
For the given coordinates \( x = -\frac{\sqrt{3}}{2} \), \( y = 0 \), and \( z = -\frac{1}{2} \), we calculate:

• First, square each value: \( (-\frac{\sqrt{3}}{2})^2 = \frac{3}{4} \), \( 0^2 = 0 \), \( (-\frac{1}{2})^2 = \frac{1}{4} \).
• Then, add these results: \( \frac{3}{4} + 0 + \frac{1}{4} = 1 \).
• Finally, take the square root to find \( r = \sqrt{1} = 1 \).

This calculation confirms the distance from the origin is exactly one unit.

Inclination Angle

The inclination angle, denoted as \( \theta \), is an important component of spherical coordinates. It indicates the angle between the positive z-axis and the line from the origin to the point. The formula used to find this angle is \( \theta = \arccos\left( \frac{z}{r} \right) \).
With \( r = 1 \) and \( z = -\frac{1}{2} \), we compute:

• Substitute the values into the formula: \( \theta = \arccos\left(-\frac{1}{2}\right) \).
• Simplifying this gives us \( \theta = \frac{2\pi}{3} \) radians.

This angle helps in accurately describing the point's position relative to the original rectangular coordinates.

Azimuthal Angle

The azimuthal angle, denoted as \( \varphi \), measures the angle in the XY-plane from the positive x-axis to the projection of the line connecting the origin to the point. Calculating \( \varphi \) involves the formula \( \varphi = \arctan\left( \frac{y}{x} \right) \).
Given our values \( x = -\frac{\sqrt{3}}{2} \) and \( y = 0 \), the calculation is straightforward:

• Using \( \varphi = \arctan\left( \frac{0}{-\frac{\sqrt{3}}{2}} \right) \), simplifies directly to \( \varphi = 0 \) radians since \( \arctan(0) = 0 \).

This angle is crucial in defining the complete orientation of the point in space within the spherical coordinate system.