Problem 66 - Advanced Engineering Mathematics

Question

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

Spherical coordinates are \((1, \pi, \frac{2\pi}{3})\).

Step 1: Identify the Rectangular Coordinates

The given rectangular coordinates in the problem are: \( x = -\frac{\sqrt{3}}{2}, y = 0, z = -\frac{1}{2} \). These are the values we'll convert to spherical coordinates.

Step 2: Calculate the Radial Distance \( r \)

The radial distance \( r \) is calculated using the formula for the magnitude of the vector: \[ r = \sqrt{x^2 + y^2 + z^2} \]Substitute the values:\[ r = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + 0^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]

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

The azimuthal angle \( \theta \) is measured from the positive x-axis in the xy-plane. Since \( y = 0 \), it simplifies the formula:\[ \theta = \arctan\left(\frac{y}{x}\right) \]However, since \( x = -\frac{\sqrt{3}}{2} \) and \( y = 0 \), \( \theta \) is either \( 0 \) or \( \pi \). Since the point is on the negative x-axis, \( \theta = \pi \).

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

The polar angle \( \phi \) is calculated from the positive z-axis:\[ \phi = \arccos\left(\frac{z}{r}\right) \]Substitute the given values: \[ \phi = \arccos\left(\frac{-\frac{1}{2}}{1}\right) = \arccos(-\frac{1}{2}) = \frac{2\pi}{3} \]

Step 5: Write the Spherical Coordinates

The spherical coordinates \( (r, \theta, \phi) \) are now summarized from our calculations:\[ (r, \theta, \phi) = (1, \pi, \frac{2\pi}{3}) \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to specify a point in a three-dimensional space using three values: x, y, and z. Think of it as a grid system, much like the one used on a map, that helps locate a place.

• The x-coordinate tells you how far to move horizontally from the origin, along the x-axis.
• The y-coordinate shows the vertical movement along the y-axis.
• The z-coordinate indicates movement up or down along the z-axis, adding depth to the position.

In our example, the given rectangular coordinates are \( x = -\frac{\sqrt{3}}{2}, y = 0, \text{ and } z = -\frac{1}{2} \). These numbers pinpoint a location in 3D space that we will convert to spherical coordinates.

Radial Distance Calculation

The radial distance, denoted as \( r \), is a measure of how far a point is from the origin, regardless of direction. It's essentially the length of the vector from the origin to the point, much like a ruler's reading tells you the length of a line.

• This distance is calculated using the formula \( r = \sqrt{x^2 + y^2 + z^2} \).
• It combines the effects of all three coordinates into a single distance.

For our coordinates \( (-\frac{\sqrt{3}}{2}, 0, -\frac{1}{2}) \), substituting into the formula gives \[ r = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + 0^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{1} = 1 \] which is quite simply the total radial distance from the origin.

Azimuthal Angle Theta

The azimuthal angle \( \theta \) is measured in the xy-plane from the positive x-axis. It is essentially like determining the direction of the point on the horizontal plane, similar to how a compass shows direction on a map.

• \( \theta \) is found using \( \theta = \arctan\left(\frac{y}{x}\right) \).
• When \( y = 0 \), this simplifies the situation considerably.

In our case, where \( y = 0 \) and \( x = -\frac{\sqrt{3}}{2} \), \( \theta \) should be either \( 0 \) or \( \pi \) (180 degrees) because these are the possible directions along the x-axis. Given that \( x \) is negative and there's no offset in y, \( \theta = \pi \), indicating the vector is in the negative x direction.

Polar Angle Phi

The polar angle \( \phi \) is taken from the positive z-axis down to the line formed by the point and the origin. Picture it like looking at how high or low the point sits in space relative to the ground.

• \( \phi \) is calculated using \( \phi = \arccos\left(\frac{z}{r}\right) \).
• It helps translate the height component of the point relative to the origin.

For our given point, substituting \( z = -\frac{1}{2} \) and \( r = 1 \) into the formula, we find \[ \phi = \arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]. This outcome indicates the angle from the z-axis to the point's position, showing us where it is vertically. This angle helps visualize the point's placement in relation to the z-axis or vertical line.