Rayan has scored 25 goals in 99 games since breaking through at Vasco da Gama
The Cherries saw off competition from a number of other European sides as well as clubs in Saudi Arabia and Russia to sign the 19-year-old Brazilian.
It is Bournemouth's joint-second biggest transfer after the £32m they paid to sign Rayan's compatriot and fellow striker Evanilson from Porto in 2024.
Rayan moves to the south coast as a replacement for attacker Antoine Semenyo, who joined Manchester City for around £65m earlier this month.
The teenager is primarily a left‑footed centre‑forward but can play on either flank. The Brazil under-20 international scored 14 goals in 34 Brazilian top‑flight matches last season.
"I am happy and honoured to be here, especially with the sporting project they developed for me," he said.
Tiago Pinto, Bournemouth's president of football operations added: "Rayan has already shown impressive consistency and maturity for his age.
"We believe he has enormous potential, and Bournemouth is the right environment for him to continue his development and make an impact. I'm excited to have him as part of our team, as we continue to reinvest in the squad."
Mandas has represented Greece at every age group from under-16
The Premier League side have also completed a loan deal for Greek goalkeeper Christos Mandas from Italian Serie A side Lazio.
The 24-year-old, who becomes the club's first Greek player, joins until the end of the season with a further option to buy.
The Cherries are paying a 1.5m euros (£1.3m) loan fee and they will need to pay an extra £860,000 if they opt against signing him on a permanent deal.
If they choose to make the move permanent, he will cost £16m.
Mandas has made 33 appearances for Lazio since joining from Greek side OFI and has won two international caps.
"When I knew Bournemouth were interested, I wanted to come here," he said.
"It feels like a magical atmosphere between the players and the fans, and I like the way the team plays, so it is the right move."
He is the second goalkeeper to move to Vitality Stadium this month following the arrival of 37-year-old former Tottenham and Celtic stopper Fraser Forster on a six-month deal.
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