Question

Explain how cylindrical coordinates are used to describe a point in \(\mathbb{R}^{3}\).

Short answer

Answer: In cylindrical coordinates, points in 3D space are represented by three parameters: radial distance (ρ), polar angle (θ), and height (z). Radial distance (ρ) represents the distance of the point from the z-axis, polar angle (θ) is the angle with the positive x-axis in the xy-plane, and height (z) is the same as in the Cartesian coordinate system.

Step-by-step solution

Understand cylindrical coordinates

Cylindrical coordinates are an alternative way to represent points in 3D space using three parameters: radial distance (ρ), polar angle (θ), and height (z). The radial distance (ρ) represents the distance of the point from the z-axis, the polar angle (θ) is the angle with the positive x-axis in the xy-plane, and height (z) is the same as in the Cartesian coordinate system.

Convert from Cartesian to cylindrical coordinates

Given a point (x, y, z) in Cartesian coordinates, we can find the corresponding cylindrical coordinates (ρ, θ, z) using the following equations: 1. ρ = \(\sqrt{x^2 + y^2}\) 2. θ = \(tan^{-1}(\frac{y}{x})\) 3. z = z

Convert from cylindrical to Cartesian coordinates

Given a point (ρ, θ, z) in cylindrical coordinates, we can find the corresponding Cartesian coordinates (x, y, z) using the following equations: 1. x = ρ * cos(θ) 2. y = ρ * sin(θ) 3. z = z

Visualizing cylindrical coordinates

To help understand cylindrical coordinates, imagine a radial line extending from the z-axis to the point in question. Then imagine a line parallel to the z-axis that intersects the point. The intersection of these two lines forms a rectangular triangle in the xy-plane. You can use the side lengths and the angle θ to determine the x and y coordinates.

Practice converting coordinates

The best way to become comfortable with cylindrical coordinates is to practice converting between Cartesian and cylindrical coordinates. Practice different combinations of points and angles to get a better understanding of how cylindrical coordinates are used to describe points in 3D space. In conclusion, cylindrical coordinates are a useful way to describe points in 3D space, particularly in problems involving cylindrical shapes or rotational symmetry. By understanding how to convert between Cartesian and cylindrical coordinates and visualizing the geometric relationships, you will be better equipped to use cylindrical coordinates in your studies.