Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(\left(\rho_{1}, \theta_{1}, \phi_{1}\right)\) and \(\left(\rho_{2}, \theta_{2}, \phi_{2}\right)\) be the spherical coordinates of two points, and let \(d\) be the straight-line distance between them. Show that $$ \begin{aligned} d=\left\\{\left(\rho_{1}-\rho_{2}\right)^{2}+2 \rho_{1} \rho_{2}\left[1-\cos \left(\theta_{1}-\theta_{2}\right) \sin \phi_{1} \sin \phi_{2}\right.\right.\\\ &\left.\left.-\cos \phi_{1} \cos \phi_{2}\right]\right\\}^{1 / 2} \end{aligned} $$

Step-by-Step Solution

Answer

The distance formula is derived by converting spherical to Cartesian coordinates and simplifying.

Step 1: Understand the Spherical Coordinates

Spherical coordinates (\(\rho, \theta, \phi\)) are used to represent the position of a point in 3D space. \(\rho\) is the radial distance, \(\theta\) is the azimuthal angle in the x-y plane from the x-axis, and \(\phi\) is the polar angle from the z-axis.

Step 2: Convert Spherical to Cartesian Coordinates

Convert both points from spherical to Cartesian coordinates. For point 1: \((x_1, y_1, z_1) = (\rho_1 \sin \phi_1 \cos \theta_1, \rho_1 \sin \phi_1 \sin \theta_1, \rho_1 \cos \phi_1)\). For point 2: \(\(x_2, y_2, z_2) = (\rho_2 \sin \phi_2 \cos \theta_2, \rho_2 \sin \phi_2 \sin \theta_2, \rho_2 \cos \phi_2)\)\).

Step 3: Use the Distance Formula in Cartesian Coordinates

Compute the distance \(d\) between the two points in Cartesian coordinates using \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]

Step 4: Substitute for Cartesian Formulas

Substitute the expressions for \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\). Calculate each coordinate difference:\[(x_2-x_1) = \rho_2 \sin \phi_2 \cos \theta_2 - \rho_1 \sin \phi_1 \cos \theta_1\]\[(y_2-y_1) = \rho_2 \sin \phi_2 \sin \theta_2 - \rho_1 \sin \phi_1 \sin \theta_1\]\[(z_2-z_1) = \rho_2 \cos \phi_2 - \rho_1 \cos \phi_1\]

Step 5: Simplify the Equation

Combine and simplify the expressions obtained into the formula for distance. Simplifying the square of the differences:\[(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 = (\rho_1^2 + \rho_2^2 - 2\rho_1\rho_2(\sin \phi_1 \sin \phi_2 \cos(\theta_1 - \theta_2) + \cos \phi_1 \cos \phi_2))\]

Step 6: Finalize Derivation

Recognize that the expression matches the requirement if you evaluate the square brackets carefully in simplified form:\[d = \sqrt{(\rho_1 - \rho_2)^2 + 2 \rho_1 \rho_2 \left[1 - \left(\cos(\theta_1 - \theta_2) \sin \phi_1 \sin \phi_2\right) - \cos \phi_1 \cos \phi_2\right]}\]

Key Concepts

Distance Formula

The distance formula is a mathematical tool used to determine the straight-line distance between two points in space. In its classic Cartesian form, it is represented as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]This formula is derived from the Pythagorean theorem, which works well for present-day geometries and has been adapted to three-dimensional space. By taking the square root of the sum of the squared differences of their respective x, y, and z coordinates, we can calculate the shortest path between the two points.

• In Cartesian coordinates, this involves straightforward subtraction of each coordinate pair, squaring the result, and then summing these squares.
• The final step includes taking the square root of the total, providing the Euclidean distance in a familiar and intuitive form.

This concept is central to various fields, including physics, engineering, and computer science, as it provides a fundamental measure of separation between points.

Cartesian Coordinates

Cartesian coordinates define a point in space using three values or coordinates corresponding to the x, y, and z-axes. They are tremendously useful because they offer a straightforward way of visualizing geometric figures and relationships between points.

• Each value represents a perpendicular distance from a central origin, offering a clear geometric interpretation of a point's position in space.
• This system was introduced by René Descartes, hence the name, and remains the backbone of much spatial analysis today.

The process of converting from spherical to Cartesian coordinates involves reimagining spatial relationships. For a point in spherical coordinates \(\rho, \theta, \phi\), its Cartesian counterparts are:\[(x, y, z) = (\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi)\]These expressions allow seamless translation between the two systems, providing a bridge for complex spatial calculations and visualizations.

Polar Angles

Polar angles come into play when working with spherical coordinates, which is a system used to locate points in a three-dimensional space. In the context of spherical coordinates, the angles provide more nuanced ways of expressing a point's location compared to Cartesian coordinates.

• The azimuthal angle, \(\theta\), represents the angle measured in the x-y plane from the positive x-axis.
• The polar angle, \(\phi\), determines the angle from the positive z-axis.

These angles, together with the radial distance, allow us to locate points not in terms of distances along axes, but rather in terms of angles and how far directly out from the origin a point is located.
This approach is especially useful in scenarios involving angles or rotational symmetry, such as astronomy or atomic physics, where understanding the angle-based position of an object or atom in space can be intuitively insightful.