I also thought it was odd that mod_alias was available, but mod_rewrite was not. I therefore contacted my hosting company and although they were not able to help in fixing the problem they did confirm that mod_rewrite was enabled.
I've therefore being playing around with the rule and I've found the problem.
Although the example code was correct for use in httpd.conf (or equivalent vhost file)
RewriteRule ^/easy$ /phpscript.php?doc=213
it is not for .htaccess. Instead the entry should have been (note the missing / at the start of the regexp).
RewriteRule ^easy$ /phpscript.php?doc=213
I suspect that this is because the .htaccess is already in the / directory it applies to rewrite rules relative to that. I did not realize that this was how it worked, I thought it was always absolute to the virtual host.
So all working correctly now. Thanks for your help.