request->to_url is similar to the url string manually created in the first approach.

As specified on vimeo oauth guide link (http://vimeo.com/api/docs/oauth), The base string should be like below:-

GET&http%3A%2F%2Fvimeo.com%2Fapi%2Frest%2Fv2%2F&method%3Dvimeo.people.getInfo%26oauth_consumer_key%3Dabcdef0123456%26oauth_nonce%3Dr4nd0m1234%26oauth_signature_method%3DHMAC-SHA1%26oauth_timestamp%3D1336052658%26oauth_version%3D1.0%26user_id%3Dbrad

There is no space in $v_search_url its a typo. (may occured when I had paste the code).