Why do so many Christians invoke this Bible verse when they feel picked on?

No Kevin it does not give us carte blanche to be rude and if a Christian does that they are wrong. However, because as Christians we believe in God's standard we will be persecuted. People who don't have a faith in Christ, don't follow the word of God. When we do we will be persecuted by those who do not think it necessary to follow His word. I beg for forgiveness for any Christian that would be hateful to someone and then quote this verse if someone is rude back to them.