If I get an IP address set up for www.my domain name.com and point an A record for www.my other domain name.com to the IP address for www.my domain name.com will going to www.my other domain name.com show the same site that www.my domain name.com shows?

(Server has Plesk 2.5)

When you are using NAMEBASED Virtualhosts in Apache, both domains will have their OWN website.

So you won't need a ip-adress a domainname

Roly i think the answer is yes if the target domain is the first created on that particular ip.