Question

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

Short answer

Question: Explain how spherical coordinates are used to describe a point in 3-dimensional space, and provide an example with a point's Cartesian coordinates given as (1, 1, 1). Answer: Spherical coordinates represent a point in 3-dimensional space using three parameters: radius (r), inclination angle (θ), and azimuthal angle (φ). The radius is the distance between a point and the origin, the inclination angle is the angle between the line connecting the center and the point and the positive z-axis, and the azimuthal angle is the angle between the projections of the line connecting the center and the point to the xy-plane and the positive x-axis. For example, consider a point P with Cartesian coordinates (1, 1, 1). To find its spherical coordinates (r, θ, φ), we can use the transformation formulas: r = √(1² + 1² + 1²) = √3 θ = arccos(1/√3) = π/4 φ = arctan(1/1) = π/4 Thus, the spherical coordinates of point P are (r, θ, φ) = (√3, π/4, π/4).

Step-by-step solution

Definition of Spherical Coordinates

Spherical coordinates are an alternative coordinate system representing a point in 3-dimensional space using three parameters: radius (\(r\)), inclination angle (\(\theta\)), and azimuthal angle (\(\phi\)). The radius is the distance between a point and the origin (i.e., the length of the vector from the origin to the point), the inclination angle is the angle between the line connecting the center and the point and the positive z-axis, and the azimuthal angle is the angle between the projections of the line connecting the center and the point to the xy-plane and the positive x-axis.

Relationship between Cartesian and Spherical Coordinates

To transform a point from spherical to Cartesian coordinates, we can use the following formulas: $$ x = r\sin\theta\cos\phi $$ $$y = r\sin\theta\sin\phi $$ $$z = r\cos\theta $$

Transformation from Cartesian to Spherical Coordinates

To transform a point from Cartesian coordinates to spherical coordinates, you can use the following formulas: $$r = \sqrt{x^2 + y^2 + z^2} $$ $$\theta = \arccos \frac{z}{\sqrt{x^2 + y^2 + z^2}} $$ $$\phi = \arctan \frac{y}{x} $$ Notice that sometimes the spherical coordinates system formally uses a range of \([0, 2\pi]\) for the inclination angle \(\theta\) and a range of \([0, \pi]\) for the azimuthal angle \(\phi\), while in other cases the conventions are reversed.

Example of Spherical Coordinates

Let's assume we have a point P with Cartesian coordinates \((x, y, z) = (1, 1, 1)\). To find its spherical coordinates \((r, \theta, \phi)\), we can use the transformation formulas: $$r = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$$ $$\theta = \arccos \left(\frac{1}{\sqrt{3}}\right) = \arccos \left(\frac{\sqrt{2}}{2} \right) = \frac{\pi}{4} $$ $$\phi = \arctan\left(\frac{1}{1}\right) = \arctan(1) = \frac{\pi}{4} $$ So the spherical coordinates of the point P are \((r, \theta, \phi) = \left(\sqrt{3}, \frac{\pi}{4}, \frac{\pi}{4}\right)\).