The drastic increase in american troop strength in vietnam in 1964 was caused directly by

Gulf of Tonkin incident.

Explanation:

The Gulf of Tonkin incident occurred after two US naval destroyers were attacked by the Vietnamese communist army in 1964. This incident led to the passage of the Gulf of Tonkin Resolution in August 7, 1964. It gave president John B Lyndon the power to take all necessary steps to retaliate any armed attack by the Vietnamese communist regime against the military of the United States. This incident led to the increase of American troop strength in Vietnam as the resolution aimed at preventing any large scale aggression or damage to the US military by Vietnam.