AMMAN, Jordan — When relatives learned Tuesday night that the Islamic State had released a video showing the death of a Jordanian fighter pilot, First Lt. Moaz al-Kasasbeh, they tried to keep it from his mother, Issaf, and his wife, Anwar. They switched off the television and tried to wrest a smartphone out of his wife’s hand, but she had already seen a mobile news bulletin.

Anwar ran crying into the street, calling her husband’s name and saying, “Please, God, let it not be true.” Issaf fell to the floor screaming, pulled her head scarf off and started tearing at her hair.

That was even before they knew how he had been killed. No one dared let them know right away that Lieutenant Kasasbeh’s tormentors had apparently burned him alive inside a cage, a killing that was soon described as the most brutal in the group’s bloody history.

Jordan responded rapidly, executing Sajida al-Rishawi, who was convicted after attempting a suicide bombing, and Ziad al-Karbouli, a top lieutenant of Al Qaeda in Iraq, before dawn on Wednesday, according to the official news agency Petra.

On Tuesday, Anwar Kasasbeh had been laughing at the memory of her husband’s delight when he discovered that her family kept rabbits in their home. After they married, her parents gave them the rabbits to take care of.

“It was so funny, he was so happy about those rabbits,” Anwar told a visiting reporter about her 26-year-old husband. “He told me how he always wanted rabbits.”

The video, with its references to the Islamic State’s punishment of nations like Jordan that joined the American-led coalition against it, appeared to be an attempt to cow the Arab nations and other countries that have agreed to battle the militants in Syria. So far, it appeared to have had the opposite effect in Jordan, which suggested its resolve had been stiffened. But the capture of the pilot had already hurt the coalition, with the United Arab Emirates suspending its own airstrikes in December and demanding that the group improve its search and rescue efforts for captured members.

Image President Obama met Tuesday with King Abdullah II of Jordan at the White House. Credit... Stephen Crowley/The New York Times

The release of the video came after weeks of growing anxiety in Jordan as the country’s leaders tried desperately to win the release of Lieutenant Kasasbeh, a member of an important tribe and the first fighter for the coalition bombing the Islamic State to be captured. Their attempts became more complicated late last month when the Islamic State, also called ISIS or ISIL, suddenly entangled the pilot’s fate with that of a Japanese man it held hostage, demanding that Jordan release Ms. Rishawi in exchange for him.

If Jordan failed to do so by last Thursday, they said, Lieutenant Kasasbeh would be killed. Jordanian officials expressed willingness to bargain, a major concession to the militants, but refused to release Ms. Rishawi until they received proof that the pilot was alive.

On Tuesday, Jordanian officials said they learned that the pilot had actually been killed on Jan. 3, suggesting that their caution had been justifiable. They did not, however, explain where they got the information.

Even by Islamic State standards, the latest propaganda video was particularly gruesome. The footage alternated images of the pilot while he was alive with segments showing the rubble of destroyed buildings and the burned bodies of Syrians allegedly killed in coalition airstrikes. Islamic State members took to Twitter to applaud the pilot’s death, calling it an eye for an eye.

At the end of the 22-minute video, an Islamic State fighter set a powder fuse alight as Lieutenant Kasasbeh watched, his clothes drenched in fuel. The flames raced into the cage and engulfed him. The camera lingered, showing close-ups of his agony, before concluding with pictures of what the Islamic State claimed were other Jordanian pilots and an offer of a reward of 100 gold coins for whoever killed one of them. (American officials said they were trying to authenticate the video.)

The Jordanian military responded swiftly. “The blood of our hero martyr, Moaz Kasasbeh, will not go for nothing,” said Mamdouh al-Ameri, a spokesman for the Jordanian military. “And the revenge will be equal to what happened to Jordan.”

Within hours, a convoy was seen leaving the women’s prison in Jordan, presumably taking Ms. Rishawi to the men’s prison an hour outside Amman where executions are carried out, normally by hanging.

Both prisoners had already been sentenced to death for terrorism offenses. Mr. Karbouli was accused as one of the planners of the 2005 hotel bombings in Amman that killed more than 57 people; Ms. Rishawi was the only one of four suicide bombers in that attack whose explosive vest failed to detonate. Both were affiliated with Al Qaeda in Iraq, which became the present-day Islamic State.

Image A new video released by Islamic State militants on Tuesday and first reported by the SITE Intelligence Group, an organization that monitors jihadist activity on the Internet, showed the execution of First Lt. Moaz al-Kasasbeh, a Jordanian air force pilot.

Jordan and the United Arab Emirates are among several Arab countries taking part in American-led air raids against Islamic State positions in Syria. Two other Arab states, plus Iraq, are members of the coalition in other capacities.

Lieutenant Kasasbeh was said to have been shot down in his F-16 fighter bomber on Dec. 24 during an air operation against Islamic State positions not far from the militants’ stronghold of Raqqa in northern Syria.

He cut a dashing figure in uniform, with green eyes, black hair and a slim build, and he had a significant social media following.

His capture transfixed the nation, which suddenly saw photos of the lieutenant being dragged by militants out of a swamp where he had apparently crashed.

Weeks before the deadly attack on the satirical newspaper Charlie Hebdo in France prompted refrains of “Je Suis Charlie,” Jordan’s Queen Rania started a campaign on Instagram called “We Are All Moaz.”

Lieutenant Kasasbeh’s captivity at first aroused anti-coalition sentiment among many in Jordan, but public opinion shifted dramatically as the Islamic State issued videos showing what it said were the beheadings of two Japanese hostages, including the one the militants had wanted to trade. By last week, critics of the coalition and the government had come under fire for trying to turn the pilot’s plight to political advantage.

For someone in the elite forefront of Jordan’s air force — its 60 or more F-16s are its most important aircraft — Lieutenant Kasasbeh did not show any early interest in the military or in flying, his family said.

“It was just by happenstance,” his father, Safi Youssef Al-Kasasbeh, said Sunday. During his last year in high school, his son, the fourth of eight children and the third son, had been planning to go to medical school in Russia, as his mother had long encouraged. But he saw a notice in a Jordanian newspaper inviting candidates to see if they qualified for the air force, and, on a lark, Lieutenant Kasasbeh applied for what would be a prestigious position.