The way to bow your head is simple. Admit that you made an excessive mistake in this quarrel, or that you really did something wrong. You should admit your mistake, and don't deny it because of self-esteem or face. It will make you lose a friend, which is really bad. The best solution to the cold war is always for one of them to bow first. Don't feel embarrassed to let yourself bow first. Actually, there is nothing to be embarrassed about. This is a sign of your maturity, and it is also a choice that mature people will make. You can make this choice for your roommate or for yourself to become more mature and sensible.
The cold war is not good. After all, they are all roommates. Frequent quarrels must hurt feelings.
But on the whole, it's nothing, because no matter how big the quarrel is, as long as you have enough time, you can pass it smoothly. Quarrel will not really affect the relationship between your. When you all wake up and recover, you will still be friends. If you really don't want to bow your head, wait for the time to pass. It will be reconciled after a while.