Thay has often said to his students, “To bow or not to bow is not the question. The important thing is to be mindful.” When we greet someone with a bow, we have the chance to be present with that person and with the nature of awake-ness, of Buddhahood, within us and within the other person. We do not bow just to be polite or diplomatic, but to recognize the miracle of being alive.
When we see a brother who joined his palms and bowed to us, we can do the same. For inspiration, we say silently "A lotus for you," during expiration, we bow and say "become Buddha." We do this by being fully aware that the person is in front of us. We bow with sincerity and with all our heart.
Sometimes we feel a very deep relationship with what is in front of us and we feel a sense of gratitude and wonder. It can be a flower, a sunset, a tree or dew drops. We bow in the same way then to offer our gratitude.