Which of the following metals having atomic number 3 is a soft, silvery metal with very low density that reacts vigorously with water and corrodes quickly in air?

Lithium

The correct answer is option 3. Lithium.

The metal with atomic number 3 that is described as soft, silvery, with very low density, reacts vigorously with water, and corrodes quickly in air is Lithium.

Lithium is a soft, silvery-white metal with a low density and is highly reactive, particularly with water and air. It is the lightest metal and the least dense solid element. When exposed to water, lithium reacts vigorously, producing hydrogen gas. Additionally, lithium readily tarnishes and corrodes in moist air. Therefore, lithium matches the description provided.