dx said:First, you have to write the equations that express x, y and z in terms of θ, φ and r. Then you write i, j and k in terms of the unit vectors [itex] e_\theta, e_\phi[/itex] and [itex] e_r [/itex]. Then you just substitute.
The formula for converting 3D cartesian coordinates (x, y, z) to polar coordinates (r, θ, φ) is:
r = √(x² + y² + z²)
θ = arccos(z/r)
φ = arctan(y/x)
3D cartesian coordinates use x, y, and z coordinates to represent a point in space, while polar coordinates use r, θ, and φ coordinates. The main difference is that polar coordinates use a distance (r) and two angles (θ and φ) to represent a point, while cartesian coordinates use three perpendicular axes.
Angles in polar coordinates have a range of 0 to 2π (or 0 to 360 degrees). This represents a full rotation around the origin point (r = 0).
Converting from cartesian to polar coordinates can be useful in certain situations, such as when working with circular or spherical objects. It can also make certain calculations, such as finding a point's distance from the origin, easier to visualize and solve.
To convert polar coordinates (r, θ, φ) to cartesian coordinates (x, y, z), use the following formula:
x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)
z = r cos(θ)