This is more a sort of "let me know if I'm right" question, rather than a real question, but I thought that if is there a place where I can find true and solid trustful answers then it's here.
Suppose someone gives me those coordinates:
$$109^{\circ} 39'\ N$$
$$30^{\circ} 10'\ W$$
If I want to localize this place over the Earth surface, via cartesian coordinates (or better: if I simply want to convert them into cartesian), here is what I understood to be the process.
Please, notice that this is what I understood so if it's wrong you have to correct me because I did not find any book or notes where this has been explained clearly.
First step
Transform the coordinates I got into "true angles" that is: latitude $\theta$ and longitude $\phi$ according to the DMS transformation:
$$\theta = 109 + \frac{39}{60} = 109.65$$
$$\phi = 30 + \frac{10}{60} = 39.16$$
Second Step
Using the polar coordinates to find the space vector, that is:
$$x = R\sin\theta\cos\phi$$
$$y = R\sin\theta\sin\phi$$
$$z = R\cos\theta$$
Where $R = $ Earth radius.
Finally I will obtain the space vector $\mathbf{r} = (x, y, z)$ for the place in question.
Final Question
Is this all right? Are there any errors or information I shall take into account? Any remark?
Thank you all.
You have a BIG mistake, which is the limit of the lattitude are $90S$ which is the South Pole to $90N$ which is the North Pole. You are here mentioning a lattitude which is 109N. This is outside the range. I assume your angles should be $109W$ and $30N$ which is somewhere Texas....
Fix your coordinates then, we can verify your transformation .