The murder of British hostage David Haines was an "act of pure evil", David Cameron has said after the release of a video of his beheading.
The 44-year-old aid worker was seized in Syria in 2013. He was being held by Islamic State militants who have already killed two US captives.
The latest video also includes a threat to kill a second British hostage.
The PM vowed to do everything possible to find the killers. Mr Haines's family said he would be "missed terribly".
Born in Holderness, East Yorkshire, Mr Haines went to school in Perth and had been living in Croatia with his second wife, who is Croatian, and their four-year-old daughter. His parents live in Ayr.
'Despicable and appalling'
In a statement released by the Foreign Office, Mike Haines said his brother, a father of two, "was and is loved by all his family".
"David was most alive and enthusiastic in his humanitarian roles. His joy and anticipation for the work he went to do in Syria is for myself and family the most important element of this whole sad affair," he said.
Mr Cameron, who chaired an emergency Cobra committee meeting on Sunday morning, said the murder of an innocent aid worker was "despicable and appalling".
"It is an act of pure evil. My heart goes out to the family of David Haines who have shown extraordinary courage and fortitude throughout this ordeal.
"We will do everything in our power to hunt down these murderers and ensure they face justice, however long it takes," the prime minister added.
'Grief and resolve'
The Foreign Office is working to verify the video, which was released on Saturday night. It begins with a clip of Mr Cameron and then features a man who appears to be Mr Haines dressed in orange overalls, kneeling in front of a masked man holding a knife.
The victim says: "My name is David Cawthorne Haines. I would like to declare that I hold you, David Cameron, entirely responsible for my execution."
He says Mr Cameron had entered into a coalition with the US against the Islamic State "just as your predecessor Tony Blair did".
"Unfortunately it is we the British public that in the end will pay the price for our parliament's selfish decisions," he said.
The militant, who appears to have a British accent, is then recorded as saying: "This British man has to pay the price for your promise, Cameron, to arm the Peshmerga against the Islamic State."
Islamic State is now in control of large parts of northern Iraq and Syria, and the CIA estimates that the group could have as many as 30,000 fighters in the region.
The UK has donated heavy machine guns and ammunition to authorities in Iraq to help fight IS militants, the Ministry of Defence has previously said.
Kurdish forces, known as the Peshmerga, have been involved in heavy fighting with IS.
Mr Haines's death has been condemned by international political leaders, including Australian Prime Minister Tony Abbott, French President Francois Hollande, Labour leader Ed Miliband and Scottish First Minister Alex Salmond.
US President Barack Obama said the US would work with the UK and other countries to "bring the perpetrators of this outrageous act to justice".
Deputy Prime Minister Nick Clegg said the government "will not rest" until Mr Haines's killers "face justice".
Mr Alex Salmond said a resilience meeting - the Scottish government's version of the UK's Cobra security meeting - would be held on Sunday morning to discuss issues such as the privacy of Mr Haines's family.
The leader of the Better Together campaign in Scotland, Alistair Darling, said Mr Haines's death "inexcusable, a barbaric act" that would "strengthen the resolve of the international community".
Justice Secretary Chris Grayling said he would like to see the militant in the video "face justice in a British court - he's a British citizen and has been boasting of that".
Analysis: Jim Muir, BBC News
The beheadings of David Haines, and the two American journalists before him, are believed to have been filmed in the desert near Raqqa, a provincial capital in north-east Syria which has become the unofficial capital of the self-styled Islamic State.
Far from being just a shadowy terror group, IS controls a large swathe of territory in the north and east of Syria, linked since June to almost all the mainly Sunni parts of Iraq.
It controls a population of around 5 or 6 million people, including Iraq's second city, Mosul.
In northern Iraq, the Kurds, helped by US air strikes, are slowly regaining the ground they lost to IS when it suddenly turned on them last month.
That's the template the Americans and their allies want to apply to other parts of Iraq, and Syria, but the situation in both countries is complicated by the absence of clear-cut, cohesive local forces on the ground to work with.
Elsewhere, religious leaders have also reacted to the killing, including the Archbishop of Canterbury, Justin Welby, who posted a call on Twitter for prayers for the family of Mr Haines, who was "evilly killed".
Shuja Shafi of the Muslim Council of Britain said: "These extremists wish to draw attention and recruits to their cause by sowing division and fear between people here in Britain. Let us deny them that luxury."
'Criminals and villains'
Colonel Richard Kemp, former commander of British forces in Afghanistan, said it was vital that Britain took some "serious action" against IS.
"The key issue here is that we cannot rule out the use of large-scale ground forces. I don't mean the kind of thing that's happening now. I mean large-scale intervention forces," he told BBC Radio 5 live.
Failure to do so would mean Britain would "just accept the fact that the Islamic State will continue to expand, continue to decapitate our citizens, continue to pose a threat to our country and countries in the region".
But former military intelligence officer Frank Ledwidge said Britain "must be sober about this, and understand our national interests and equate what we do with those interests".
He said IS's objective now is "to hold what it's got, and there's no better way of doing that than recruiting to it the millions of people in the area who'd be quite happy to fight yet another Western intervention in the area".
Paul Rogers, professor of peace studies at Bradford University, said the UK's natural reaction would be to "go in hard" against IS, but that this presented a problem, because the extremists wanted to be attacked so as to "present itself as fighting the far enemy".
Mr Rogers also said the "war" in the region was "at a much higher level than we suppose", with more than 2,000 sorties by the US in the last few weeks.
The West was now moving into "what is essentially a third Iraq war" and that this time it would extend into Syria, he said.
Mr Haines was taken hostage in the village of Atmeh, in the Idlib province of Syria, in March 2013.
He had been helping French agency Acted deliver humanitarian aid, having previously helped local people in Libya and South Sudan.
The release of the video came hours after his family made a direct appeal to IS to contact them.
IS - also known as Isis or Isil - has seized large swathes of territory in Iraq and Syria and declared a new caliphate - or Islamic state.
Militants from the extremist group have killed two US hostages in recent weeks, posting video evidence on the internet.
They threatened to kill Mr Haines during a video posted online showing the killing of US journalist Steven Sotloff earlier this month.
The extremist group also killed fellow US journalist James Foley last month.