The way that leaders are elected and rules established in both Houses of Congress, goes like this: The first order of buiness for both Houses in the new Congress in January, and every two years as long as we have a Congress, is choice of leaders and writing of Rules.
First, the Parties caucus separately, and make their own choices. However fractious the inter-Party fighting may be in those caucuses, all members of both Parties are expected to follow the "unit rule" on the floor. That means ALL agree to support the choice of their Caucus. The result is, the Party with the most members chooses the leadership and writes the Rules.
Since the Senate has an even number of members (unlike the House), it can result in a tie vote. Though the Vice President is NOT a member of the Caucus of his Party, he DOES vote on the floor if there is a tie. So, if the Senate is divided 50-50 in January (counting Benedict Jeffords as a Democrat), Cheney will cast the deciding vote.
Read the biography of John Adams. He cast perhaps the maximum number of tie-breaking votes in history as Vice President under George Washington.
Congressman Billybob
Click for "Oedipus and the Democrats"