Cylindrical and Spherical Coordinates Changing

[theBEAST]

Homework Statement

Convert the following as indicated:

1. r = 3, θ = -π/6, φ = -1 to cylindrical

2. r = 3, θ = -π/6, φ = -1 to cartesian

The Attempt at a Solution

I just want to check if my answers are correct.

1. (2.52, -π/6, 1.62)
2. (-2.18, -1.26, 1.62)

 

[LCKurtz]

Homework Statement

Convert the following as indicated:

1. r = 3, θ = -π/6, φ = -1 to cylindrical

2. r = 3, θ = -π/6, φ = -1 to cartesian

The Attempt at a Solution

I just want to check if my answers are correct.

1. (2.52, -π/6, 1.62)
2. (-2.18, -1.26, 1.62)

What coordinate system are these given in? $r$ is usually used in cylindrical coordinates and $\rho$ for spherical. Also, if $\phi$ is the spherical coordinate angle from the $z$ axis, it is usually restricted to the interval $[0,\pi]$. Are you sure you copied the $\phi$ values correctly?

 

[vela]

Mathematicians and physicists use $\theta$ and $\phi$ differently. You need to tell us which convention you're using here.

 

[theBEAST]

Here is the question with the answer key:

In this case r = ρ and I'm not sure why phi is negative.

I don't think the answer key is correct.

 

[vela]

The answer key is correct. You need to show us your calculations. To answer #1, it's probably most straightforward if you do #2 first and the convert from Cartesian to cylindrical.

 

[theBEAST]

The answer key is correct. You need to show us your calculations. To answer #1, it's probably most straightforward if you do #2 first and the convert from Cartesian to cylindrical.

For number one, how can r be negative? They have -2.52 whereas I have 2.52. It is why I thought the answer key was wrong.

 

[vela]

When r is negative, you reflect through the origin from where you'd otherwise be. In polar coordinates, for instance, the point r=-1, θ=π/4 would correspond to (-1/√2, -1/√2), which is where you'd end up if you reflected r=1, θ=π/4 through (0,0).

You'll notice they gave you a second answer where r is positive, but the angle has been changed to account for the reflection.

 

[theBEAST]

When r is negative, you reflect through the origin from where you'd otherwise be. In polar coordinates, for instance, the point r=-1, θ=π/4 would correspond to (-1/√2, -1/√2), which is where you'd end up if you reflected r=1, θ=π/4 through (0,0).

You'll notice they gave you a second answer where r is positive, but the angle has been changed to account for the reflection.

Lastly, when ρ is negative does that mean the angle starts from the -z axis? Because when ρ is positive it starts from the positive z axis.

 

[vela]

Not exactly. Reflection in spherical coordinates takes $\phi \to \pi-\phi$ and $\theta \to \theta+\pi$. The change to $\phi$ effectively means you're measuring from the -z-axis, but you also have to accompany it with a rotation by 180 degrees about the z-axis.