Ferdinand announced on Monday his career with the Red Devils had come to an end, following centre-back partner Vidic out of the club as the Serbian had already signed a deal with Inter Milan for next season.
The defensive pair were pivotal to United's success in recent years and they were integral parts of the title-winning teams in 2007, 2008 and 2009 - both were named in the PFA Team of the Year at the end of each campaign - and Carrick knows they will be missed.
1st Vida now Rio, we've shared so much.. Both 2 of the very best defenders of ALL TIME and I don't say that lightly, #ThePerfectPartnership"
"Vida, simply the best out and out defender I've seen. Aggression and desire to win at all costs, scared centre forwards to death!
"Rio, A Rolls Royce, best all round CB, got the lot, defend, quick, composed beyond belief and made those around him better players."
Carrick was joined in his appreciation of Ferdinand by current and former goalkeepers David de Gea and Edwin van der Sar.
"Thanks for everything @rioferdy5 and good luck in your future my friend. It's been a pleasure playing and learning with you! #Legend,".
"Just want to thank @rioferdy5 & Vida for all they have done for me in the success we achieved @ManUtd!#rocksolid."
The decision to let Ferdinand go ignores the advice of interim manager Ryan Giggs, who insisted earlier this month the defender could still make a useful contribution and should be offered a new deal.
Former Reds defender Gary Neville appeared to question the move when he wrote on Twitter: "So it looks like Rio and Vida allowed to leave at once!!! A conveyor belt has become a cliff!"
Ex-United striker Dwight Yorke described the 35-year-old Ferdinand as "colossal".
"The word colossal comes into play as that's the type of player that Ferdinand was," he told Sky Sports.
"It doesn't come as a surprise but it's an end of an era for himself and Nemanja Vidic - it's unbelievable the amount of success they've had.
"Rio can leave that football club with his head held incredibly high in terms of what he has achieved."
Ferdinand moved to Old Trafford from Leeds in July 2002 for a then British record transfer fee of £29.1million, which also made him the world's most expensive defender, and he went on to win the Premier League six times, the Champions League once, two League Cups and a Club World Cup.
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