How do I compute a vector given Right Ascension, Declenation?

Question

So, I want to calculate the vector between two points on Earth. I know the Right Ascension/Declination of a telescope’s beam-pointing-center when posting at a distant star for both points on earth. I want to subtract those vectors to product the vector between the observers (https://pasteboard.co/JhJBFm9.png). Is there a formula for computing a vector given RA/DeC?

Answer 1

You can first use spherical coordinates to find the position of the distant star relative to both stations, provided you have the distance to the star. In your case, the right ascension is equal to $\theta$, the distance to the star is $r$, and $90º - \text{declination}$ is equal to $\phi$ (due to the different conventions in mathematics).

Then convert to Cartesian coordinates using the formulas:

$$x = r \cos \theta \sin \phi, y = r \sin \theta \sin \phi, z = r \cos \phi.$$

Once you know two of the three vectors, the third vector which is between the two points on Earth can be found by subtracting the vectors. This vector will pass through Earth's interior, so if you want to find the (surface) distance between the two points, you need another method.