In the ancient Near East, lying face down in the dust showed submission (Genesis 17:1–3; Leviticus 9:24; Joshua 7:6; 1 Samuel 5:4; 1 Kings 18:39; 1 Chronicles 21:16; Matthew 17:5–6).
To turn the other cheek shows a willingness to submit rather than fight back. Jesus likely had this verse in mind when he taught his disciples to accept persecution (Matthew 5:39).
This response shows trust that God rules over all things, even over suffering.