Final answer:
This question deals with mathematical coordinate systems, focusing on the conversion between polar and rectangular coordinates, and extending to spherical coordinates for three-dimensional representation. Using polar coordinates simplifies problems involving symmetry around a point, and spherical coordinates further extend this to three-dimensional space.
Step-by-step explanation:
The question provided pertains to the concept of coordinate systems in mathematics, specifically comparing polar coordinates and rectangular (or Cartesian) coordinates. Polar coordinates define a point in a plane using the distance from a reference point (the origin) and the angle from a reference direction (usually the positive x-axis). They are typically used in situations involving circular and spherical symmetry, where they offer a more natural representation than rectangular coordinates.
In polar coordinates, a point is described by (r, θ), where r is the radial distance from the origin and θ is the angle measured from the positive x-axis. The rectangular coordinates (x, y) can be obtained using the formulas x = r × cos(θ) and y = r × sin(θ).
Spherical coordinates, on the other hand, are an extension of polar coordinates into three dimensions and are used to represent points in space. A point in spherical coordinates is described by (r, θ, ϕ), where r is the radial distance, θ is the polar angle measured from the positive z-axis, and ϕ is the azimuthal angle in the xy-plane from the positive x-axis. The conversion to rectangular coordinates (x, y, z) is given by x = r × sin(θ) × cos(ϕ), y = r × sin(θ) × sin(ϕ), and z = r × cos(θ).