The US Coast Guard has confirmed it has restarted the search for the four missing UK sailors whose yacht capsized in the Atlantic Ocean.
US Coast Guard petty officer Jennifer Robertson said: "The search has resumed."
Dan Carpenter, son-in-law of Steve Warren, one of the missing men, said: "We are holding out hope. We are aware that it is still a long shot but while there is some hope, we are concentrating on that."
Prime Minister David Cameron said: "My thanks to the US Coastguard, which has resumed its search for our missing yachtsmen."
The crew of the 40ft (12m) Cheeki Rafiki ran into difficulties about 620 miles (998km) east of Cape Cod in Massachusetts last Thursday while returning to the UK from a regatta in Antigua. Contact with the yacht was lost in the early hours of Friday when they diverted to the Azores.
The coastguard, Canadian aircraft and three merchant vessels searched for them throughout Friday and Saturday but called off efforts on Sunday at 5am local time amid treacherous weather.
Relatives of the four men - experienced captain Andrew Bridge, 22, and crew members James Male, 23, Steve Warren, 52, and Paul Goslin, 56 - have been pleading with the US Coastguard to resume the search and remain convinced that their loved ones are alive.
Andrew Bridge's grandmother Valerie said: "We are delighted. It is at least something and that is all we were asking for, all we wanted was another search.
"It might not come to anything but people want them to do it and they are trying. It seemed too quick, just two days and we were saying 'if only they could do it (search) for a bit longer'. You never know what could happen."
The families of the sailors have travelled to London to meet foreign minister Hugh Robertson.
Leading figures from the yachting world are among more than 175,000 people who have signed an online petition supporting the campaign to resume the search.
Health Secretary Jeremy Hunt, the local MP for one of the missing men, had appealed to the US Coast Guard (USCG) not to give up and entrepreneur and adventurer Sir Richard Branson had called on vessels near the area to keep a lookout.
Dame Ellen MacArthur, who twice broke the world record for fastest solo circumnavigation of the globe, said there was "every chance" that the sailors were still alive.
She said: "It seems that there is an element of everyone working in the dark here as the incident happened so far from land and there is no longer contact with the crew. However there is every chance that the sailors could be alive either inside the hull of the vessel, or in the life-raft which is designed to keep people alive at sea.
"There are examples of both types of survival, and in both cases for extended periods of time."
Mike Golding, one of the few yachtsmen to have sailed around the world non-stop in both directions, had argued for a further search, especially to locate the hull and establish whether a life raft had been released.
He said: "It's hard to imagine four men, well-equipped, even with a boat in trouble actually disappearing like that.
"The boat had all the right safety equipment, people are just saying the search period was very short."
Some 4,000 square miles (10,360 sq km) were scanned for the vessel's two personal location GPS beacons until no more transmissions were received from the small devices, which have a short battery life.
On Saturday, a cargo vessel which was helping with the search spotted and photographed an overturned hull which matched the description of the Cheeki Rafiki but reported no signs of people on board or a life raft.
Mr Golding said: "From the images, the yacht has lost the keel, initially they were sinking, taking on water. One imagines they put out the Mayday, prepared themselves for sinking, then the keel fell off, maybe the boat rolled over fast and the question is what happened at that point? Were they able to launch the life raft at that point?"
He added that efforts should be made to locate the hull, which he believed could still be afloat, having lost its heavy keel, to establish whether the life raft was still in place and if the sailors were gathered in an air pocket underneath.
Graham Male, from Romsey, Hampshire, the father of crew member James Male, said: "We are absolutely over the moon.
"We haven't got the full details yet as we are in the process of travelling to the Foreign Office to receive those details.
"From what we have heard it's really what we have been campaigning for."
Mr Male said he wished to thank everyone who had supported the online petition and added: "This is what we wanted, the public have been behind us, the celebrities have been behind us, the RYA (Royal Yachting Association) have been behind us, all the expert opinions, people who have been in this situation before."
He said that he was optimistic that they would now find out what has happened to the Cheeki Rafiki and added: "We have to get some kind of resolution now."
Nicola Evans, Mr Bridge's friend who started the petition, said: "We have just heard that the US Coast Guard will be resuming the search for the Cheeki Rafiki crew. This is amazing. I'm overwhelmed.
"When we started this campaign, we didn't know who would listen, just that our boys were lost at sea and we refused to let go of hope."
She said that everyone is " so grateful" to the US Coast Guard for listening to all the people who had backed the petition. By 3.30pm today more than 197,200 had signed it.
She said: "Thank you again, so much, for helping my amazing friend Andy and his crewmates by giving them a fighting chance. The impact you have made by signing this petition is unbelievable."
Sir Robin Knox-Johnston, the first man to sail non-stop round the world, said: " The US Coast Guard and US Navy are the best at this, they have the most experience and they have the assets.
"I feel that we humans are better at surviving than is often reflected in statistics, so I think there should be just one more sweep downwind of where the hull was discovered, so that people can feel that everything that could be done has been done. Our thoughts are with the families of the crew."
Four-time Olympic champion Sir Ben Ainslie, the world's most successful Olympic sailor, told BBC News: "If there is a chance they are still out there, then we need to keep looking for them, so it is fantastic news that the US Coast Guard have agreed to get back out there and they should be commended for that."
He felt it was a moment when families and loved ones of the missing men might be able to think that " everything that possibly can be done is being done".
Sir Ben also noted that it had come after a " huge response" which had come nationally, internationally and from the local sailing community in the UK to continue searching for the sailors.
The families of the yachtsmen will travel to the US Embassy to meet the homeland attache after their meeting at the Foreign Office, Mr Goslin's niece Gemma Townsend said.
Ms Townsend said the families would be offering their thanks for the resumption of the search. She said she did not know the name of the person due to meet them.