Final answer:
A 'search and destroy' mission is an offensive task utilized notably by US troops during the Vietnam War to eliminate hostile forces by seeking out and destroying them and their supply lines.
Step-by-step explanation:
The offensive task designed to catch or cut off a hostile force attempting to escape, with the aim of destroying it, is known as a search and destroy mission. This method was notably employed by US troops during the Vietnam War to counter the guerrilla tactics of the Vietcong and North Vietnamese forces. The objective of these missions was to seek out the enemy and their supply lines, often through deployment via helicopter, and to destroy them, thereby disrupting their operations and capabilities. Such strategies are consistent with historical military tactics that emphasize outmaneuvering the enemy, as seen in the advice to avoid direct frontal attacks and to instead find ways to flank or surprise the enemy, creating opportunities to strike effectively. However, these tactics also imply a deep understanding of the enemy's movements and psychology, as outlined in classic texts like Sun Tzu's The Art of War, which advises to capitalize on the enemy's movements and to deceive and ambush when the moment is right.