- #1
xoxomae
- 23
- 1
Is the position vector r=xi+yj+zk just r=rerin spherical coordinates?
pixel said:
A position vector in spherical coordinates is a mathematical representation of a point in three-dimensional space using three coordinates: radius (r), inclination (θ), and azimuth (φ). It is used to describe the location of a point relative to a fixed origin.
In Cartesian coordinates, a position vector is represented as (x, y, z), where x, y, and z are the distances from the origin in the x, y, and z directions respectively. In spherical coordinates, the position vector is represented as (r, θ, φ), where r is the distance from the origin, θ is the angle from the positive z-axis, and φ is the angle from the positive x-axis. So, while both systems use three coordinates, the way they are measured and represented is different.
To convert a position vector in spherical coordinates to Cartesian coordinates, we use the following formulas:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
Here, r is the distance from the origin, θ is the angle from the positive z-axis, and φ is the angle from the positive x-axis.
The range of values for the coordinates in a position vector in spherical coordinates are as follows:
r: r ≥ 0 (r cannot be negative as it represents a distance)
θ: 0 ≤ θ ≤ π (θ is the inclination angle, so it ranges from 0 to 180 degrees)
φ: 0 ≤ φ ≤ 2π (φ is the azimuth angle, so it ranges from 0 to 360 degrees)
Position vectors in spherical coordinates are commonly used in fields such as astronomy, physics, and engineering. They are especially useful in situations where the location of a point needs to be described in a spherical system, such as the position of planets, stars, or satellites in space.