Transforming Coordinates: From Rectangular to Spherical

In the field of mathematics, the ability to navigate between different coordinate systems is a crucial skill. One such transformation is from rectangular (Cartesian) coordinates to spherical coordinates. This article will guide you through the process of converting two specific points from rectangular to spherical coordinates, providing a detailed explanation of the underlying concepts.

Understanding Rectangular and Spherical Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most commonly used coordinate system. In a three-dimensional space, a point is represented by the triplet (x, y, z), where x, y, and z are the distances from the origin along the respective axes.

Spherical coordinates, on the other hand, describe a point in space using a radial distance (r) from the origin, an angle (θ) measured from the positive x-axis in the xy-plane, and an angle (φ) measured from the positive z-axis. This coordinate system is particularly useful for representing points on the surface of a sphere or in spherical-symmetric problems.

Converting Coordinates: Examples

Let's consider two specific points and transform them from rectangular to spherical coordinates.

Example 1: (3, 3√3, 6√3)

To convert the point (3, 3√3, 6√3) from rectangular to spherical coordinates, we need to follow these steps:

1. Calculate the radial distance (r): r = √(x^2 + y^2 + z^2) r = √(3^2 + (3√3)^2 + (6√3)^2) r = √(9 + 27 + 54) r = √90 r ≈ 9.49

2. Calculate the angle (θ) in the xy-plane: θ = tan^-1(y/x) θ = tan^-1(3√3/3) θ = tan^-1(√3) θ ≈ 60°

3. Calculate the angle (φ) from the positive z-axis: φ = tan^-1(√(x^2 + y^2)/z) φ = tan^-1(√(3^2 + (3√3)^2)/6√3) φ = tan^-1(√(9 + 27)/18) φ ≈ 53.13°

Therefore, the point (3, 3√3, 6√3) in rectangular coordinates corresponds to the spherical coordinates (r ≈ 9.49, θ ≈ 60°, φ ≈ 53.13°).

Example 2: (0, 5, 5)

Now, let's convert the point (0, 5, 5) from rectangular to spherical coordinates.

1. Calculate the radial distance (r): r = √(x^2 + y^2 + z^2) r = √(0^2 + 5^2 + 5^2) r = √50 r ≈ 7.07

2. Calculate the angle (θ) in the xy-plane: θ = tan^-1(y/x) θ = tan^-1(5/0) θ = 90°

3. Calculate the angle (φ) from the positive z-axis: φ = tan^-1(√(x^2 + y^2)/z) φ = tan^-1(√(0^2 + 5^2)/5) φ = tan^-1(1) φ = 45°

Therefore, the point (0, 5, 5) in rectangular coordinates corresponds to the spherical coordinates (r ≈ 7.07, θ = 90°, φ = 45°).

By understanding the relationships between rectangular and spherical coordinates, you can confidently convert points between these two coordinate systems, which is a valuable skill in various mathematical and scientific applications.