Westminster attacker Khalid Masood acted alone and there is no information to suggest further attacks are planned, UK metropolitan police say.
Deputy assistant commissioner Neil Basu said: "We must all accept that there is a possibility we will never understand why he did this."
Four people died and 50 were injured when Masood drove his car into people before stabbing a police officer.
The family of PC Keith Palmer said his bravery will be remembered.
MP Tobias Ellwood, who was among those who tried to save PC Palmer's life, has said he is "heartbroken" that his efforts were not successful.
Detectives confirmed the attack was over within 82 seconds.
Mr Basu said: "We still believe that Masood acted alone on the day and there is no information or intelligence to suggest there are further attacks planned.
"Even if he acted alone in the preparation, we need to establish with absolute clarity why he did these unspeakable acts to bring reassurance to Londoners, and to provide answers and closure for the families of those killed and the victims and survivors of this atrocity.
"Nevertheless, we are determined to understand if Masood was a lone actor inspired by terrorist propaganda or if others have encouraged, supported or directed him.
"If the latter proves to be the case, they will face justice."
He urged those who knew Masood to speak to police.
In a statement released on Saturday, the family of PC Palmer addressed those who tried to save his life.
"There was nothing more you could have done," it said.
"You did your best and we are just grateful he was not alone.
"We care about him being remembered for his selfless bravery and loving nature. We miss him so much, but we are also incredibly proud of Keith."
The family said they had been "overwhelmed by the love and support" shown for them and for PC Palmer and they praised the support from the police.
In a statement, also released on Saturday, Mr Ellwood said: "I am heartbroken that I could not do more for PC Keith Palmer who gave his life in holding the line against terrorism and defending democracy."
The Foreign Office minister said he was "deeply humbled and overwhelmed" by messages of support.
"I played only a small part that day, doing what I was taught to do, and am honoured to have been invited to join the Privy Council afterwards," he said.
"It is right that we concentrate our thoughts on the victims as we stand side by side to protect all that we hold dear, including our precious values and way of life which will always prevail."
Masood's other victims were Aysha Frade, who was in her 40s and worked at a London sixth-form college, US tourist Kurt Cochran, 54, and retired window cleaner Leslie Rhodes, 75, from south London.
Nine people released without charge
A 58-year-old man, who was arrested in Birmingham the morning after the attack under the Terrorism Act, remains in custody, and a 32-year-old woman who was arrested in Manchester, remains on police bail, Metropolitan Police have said.
Eleven people were initially arrested over the incident and nine people in total have been released without charge.
The Metropolitan Police said Masood, 52, who had previous criminal convictions but none for terrorism, had used a number of aliases.
At birth, he was registered in Dartford, Kent, as Adrian Elms, but later took his stepfather's name becoming Adrian Ajao in childhood.
In the early 2000s, he was convicted of causing grievous bodily harm after slashing a man across the face with a knife in a pub.