It's a better idea to use sub*domains* for this, just create another A-record for the subdomain and then you're done.
If you want to use the sub-directories still, you have two options:
Apache's mod_rewrite
to write a subdomain to a directory, where you'd create sub2.example.com
and it would be accessible via example.com/dir2
My recommended approach, however, is to use the mod_proxy
(Make sure all of the mod_proxy* modules are enabled).
<VirtualHost *:80>
ServerAdmin webmaster@localhost
<Location /directory2>
ProxyPass http://remote-ip-addr:80/
#you can have servers running on any port
Order allow,deny
Allow from all
</Location>
</VirtualHost>