On what basis did the majority of court justices find school prayer unconstitutional

The majority of court justices in the case Engel v. Vitale case found school prayer to be unconstitutional because it violates the Establishment Clause of the First Amendment. The First Amendment has two clauses. One of these, the Establishment Clause, prohibits the government from "establishing" a religion. That is, the government cannot set up a church or create laws that aid a particular religion or even every single religion in existence. The prayer that students were reciting (though voluntarily) was created by a government official and having students recite it in a public school (even voluntarily) was considered in opposition to this clause by the majority of justices who heard the case. Another way to understand this clause is to think of it is part of the legal work that fosters a separation of church and state.

I'm thinking that we're talking here about the court case Engel v. Vitale, from 1962. A New York state law required each school day to start with the Pledge of Allegiance and a nondenominational prayer. Students were allowed to refrain from the prayer if they found it objectionable. But the court decision found that the establishment of the prayer as a standard part of the school day in a public school was in violation of the First Amendment to the Constitution, which prohibits the government from making any law that establishes a religion.