Correct Answer: waiting at the airport

Solution : The correct choice is the first option.

Explanation : The error lies in the incomplete expression of the continuous action. The present participle "waiting" should be accompanied by a helping verb to form the correct present perfect continuous tense. The sentence should be revised to "The tired and vexed travellers have been waiting at the airport for a long time." This correction ensures proper grammatical structure by incorporating the helping verb "have been" to convey the ongoing nature of the waiting.

So, the correct sentence would be: The tired and vexed travellers have been waiting at the airport for a long time.