What condition would allow the Vice President to vote in the Senate?

The Vice President of the United States is given the authority to cast a vote in the Senate in the event of a tie. This is established by the Constitution, which specifies that the Vice President serves as the President of the Senate and has the power to break ties. Thus, when the vote among senators results in an equal split, it becomes essential for the Vice President to step in and cast the deciding vote.

In contrast, the other situations would not grant the Vice President the right to vote. The absence of a majority of senators does not create a scenario that permits the Vice President to vote; rather, it may result in a lack of quorum. Additionally, being directed by the president does not inherently give the Vice President the opportunity to vote on Senate matters unless there is a formal tie. Lastly, even if a motion is made for participation, the Constitution clearly delineates the circumstances under which the Vice President can vote, reinforcing that the decisive moment is limited to tie-breaking.