I am trying to connect my sony Vaio laptop with HDMI output to a digital projector which is VGA (1024x768). Is this even possible with just a cable? I tried this already with no connectivity issues.
Do NOT get the regular cable, you need an active converter, the first link is the right one. You need to have specific hardware to use one of those cables without a converter. Read the full description on the cable it will tell you what it can be used with.
The other thing, you said you tried this already with no issues, if you have no issues why are you asking?