How did the Gulf of Tonkin Resolution give the president a free hand?
The Gulf of Tonkin Resolution gave the president a completely free hand with regard to the Vietnam War. It gave him the ability to essentially do whatever he wanted in that war. He would no longer have to get Congressional approval to do anything like sending more troops to Vietnam.
The resolution gave the president many powers. It allowed him "to take all necessary measures" to protect US troops from attack and to prevent further attacks. It allowed him to "take all necessary steps" to help any country in southeast Asia. Finally, it gave the president the right to decide (though Congress could decide sooner if it wanted to) when the resolution expired.
In these ways, the resolution gave the president a free hand. It explicitly gave him permission to do whatever he felt necessary to win the war without having to consult with Congress to do so.