What kind of vow did Paul take in Acts 21:23?

Paul visited James in Jerusalem. James knew that troubles would come from the Jews against Paul. They incorrectly claimed that Paul was destroying Moses' Law. James and the brothers came up with a plan to pacify the Jews in Acts 21:

23 Do therefore what we tell you. We have four men who are under a vow;

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What kind of vow?

24 Take these men and purify yourself along with them and pay their expenses, so that they may shave their heads.

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Nazarite vow, Numbers 6:18

Look, Paul and his men were following Moses' law even the Nizirtie vow.

Thus all will know that there is nothing in what they have been told about you, but that you yourself also live in observance of the law. ...

26Then Paul took the men, and the next day he purified himself along with them and went into the temple, giving notice when the days of purification would be fulfilled and the offering presented for each one of them.

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Did the plan work?

No.

27 When the seven days were almost completed, the Jews from Asia, seeing him in the temple, stirred up the whole crowd and laid hands on him, 28 crying out, “Men of Israel, help! This is the man who is teaching everyone everywhere against the people and the law and this place. Moreover, he even brought Greeks into the temple and has defiled this holy place.”

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People often are not reasonable. They just wanted to believe what they wanted to believe.