Is it considered a breach of an oath (hanth) in a divorce vow ("By divorce, I will not ride with him again") if the swearer rides with the owner of the first car in his own private car after purchasing a car, or if they meet in a third person's car, a bus, or an airplane, or if he extends his hand to shake hands inside the car of the oath-taker and the car moves before he withdraws his hand?

The basis of an oath is the intention of the one swearing it. If the swearer intended not to meet with the person in any mode of transport, then he breaks his oath if a single vehicle brings them together. However, if he intended not to ride specifically in his car, then he does not break his oath unless he rides in his car. The foundation of an oath is built upon the swearer's intention. If he intends by his oath something that it can bear, then his oath is directed by that intention, whether it conforms to the wording or contradicts it. And greeting someone inside the car does not necessitate breaking the oath.