To find the angular resolution of the Very Long Baseline Interferometer (VLBI), we can use the formula resolution = wavelength / baseline. For the VLBI, the wavelength is about 1 cm and the baseline is approximately the Earth's diameter, or about 12742 km. We first convert these measurements into a matching scale, in this case, we reduce baseline to centimeters and convert wavelength to kilometers, then perform the division to get resolution in radians. There is 1 radian equal to 206265 arcseconds, so we convert the resolution from radians to arcseconds to get an angular resolution of 1.6187804112384242e-07 arcseconds.
Next, to find the angular sizes of the Sun, Jupiter, and a Sun-like star 10 light years away, we use the formula angular size = object size / distance.
For the Sun, the size is approximately 1.3914e6 km, and the distance from Earth is about 1.496e8 km. We convert these measurements into a matching scale, in this case kilometers, and then perform the division to get the Sun's angular size in radians. Converting from radians to arcseconds gives us an angular size of 1918.429953208556 arcseconds.
For Jupiter, the size is about 1.42984e5 km, and the distance to Earth is about 7.786e8 km. Using the same method as above, the angular size is found to be 37.87900688415104 arcseconds.
For a Sun-like star 10 light years away, we assume it's about the same size as the Sun, so 1.3914e6 km, and the distance is 10 light years, or about 10 * 9.461e12 km. Doing the same calculation as above yields an angular size of 0.003033475541697495 arcseconds.
To sum up, the VLBI angular resolution is much smaller than the angular sizes of the Sun, Jupiter, or a Sun-like star 10 light years away. This means that the VLBI has excellent resolution and can distinguish between these celestial bodies despite their great distances from Earth.