Are vows made in the mind accompanied by a nod of the head or saying "yes," or uttering "I promise, O Lord" with the intention of fulfilling what is in mind, or swearing by Allah with the intention of fulfilling the vow that is in mind, considered binding vows? And is it obligatory to fulfill a vow of charity if it was not made with the intention of drawing closer to Allah?

A vow (nadhr) is not binding unless expressed in a form that indicates commitment. What a person intends in their heart is not considered a vow, even if they utter the word "yes" or nod their head to indicate agreement. The statement "I promise, O Lord, I will fulfill what I intended" is not a vow; rather, it is a promise that is commendable to fulfill but not obligatory according to the majority of scholars. As for swearing to fulfill what one intended, that is an oath, and if one does not fulfill it, an expiation for breaking an oath becomes binding upon them. It is not a condition for the obligation of fulfilling a vow to have intended closeness to Allah; rather, whenever one vows to perform an act of obedience that was not obligatory upon them before the vow, it becomes obligatory to fulfill it.