Falcao joined United in a surprise deadline day loan move from Monaco and will play for the club until at least the end of the Premier League season.
Van Gaal has been tasked with reviving United's fortunes after a disastrous seventh-place finish last term and has splashed the cash in a bid to return the club to its former glories.
Angel Di Maria, Luke Shaw, Ander Herrera, Marcos Rojo and Daley Blind all joined Falcao as new United arrivals in the recently closed transfer window.
The club's spending was estimated at over £150 million as a result and the acquisition of Falcao is a significant coup, adding further depth to a forward line already containing proven Premier League stars Wayne Rooney and Robin van Persie.
Falcao is keen on a long-term stay at United and is confident that he will hit the ground running at the club, who he is tipping to bounce back from a poor start in strong fashion.
The Colombia striker told a news conference on Thursday: "I am very happy to be here. It's a big challenge for me, for my career.
"I always dream to be in a club like Manchester United. This is a new process, a new team.
"This year some players [come] and I think that Manchester United build a big team this season and in future season. I will like to be here and to be part of this project.
"I [don't] think I will need too much time [to adapt] because Manchester United has very good players.
"And that reason is easier to play football. I'm used to life in cities like this. I don't have [any] problem for adaptation."
Falcao has made just three club appearances since injuring his knee in January but asked if it felt as good as new, he responded: "Yes. I feel well. I started to play two months ago with Monaco.
"And I'm improve [sic] in the last month a lot. I score goals. That is important to strikers. I am confident in my physical form. I am comfortable with my knee."
Falcao was flanked by Blind and manager Van Gaal and the Netherlands utility man is delighted to link up with his former coach at national level.
Blind is hopeful his versatility will make him an Old Trafford regular.
"It's really important for me," Blind said.
"He gave me a lot of trust during the World Cup and the qualification games for the Dutch team. I'm happy to work again with the coach. I learn a lot from him and I hope I learn more.
"For me, it's important that I can play in both positions, as defender and midfielder. I want to play."
