what is the background situation of Paul's vow ? (Acts 18:18) After this, Paul stayed many days longer and then took leave of the brothers and set sail for Syria, and with him Priscilla and Aquila. At Cenchreae he had cut his hair, for he was under a vow. (Acts 18:18, ESV)
A theme repeated consistently throughout Acts of the Apostles is that the Jews unfairly attack Paul for not following Jewish religious laws. In each case, Acts defends Paul by providing evidence that he was a loyal Jew, even if the defence sometimes seems contrary to what Paul states in his epistles to be his beliefs, including that Christians are not under Jewish law.

The vow in Acts 18:18 that required Paul to cut his hair would be the Nazarite vow that is stipulated in Numbers 6:1-21.  Referring to claims that Paul taught his followers to abandon the Mosaic laws, Dennis E. Smith and Joseph Tyson say, in Acts and Christian Beginnings, page 225, "Luke intends to show that this is a false charge, and the taking of a vow of special devotion is probably intended to support the image of Paul as a faithful Jew."