Is it permissible to go out with one's mother in the car if she is accompanied by a woman who is immodestly dressed (mutabarrijah) or scantily clothed ('aasiyah 'ariyah)?

Obeying one's mother is obligatory as long as she does not command disobedience. The Shari'ah has commanded lowering the gaze from what is unlawful. Accompanying a woman who displays her adornment makes it difficult to fulfill the command of lowering the gaze, and avoiding the forbidden is obligatory. Therefore, it is not permissible to obey one's mother in going on a trip where a woman who displays her adornment accompanies you, if you are unable to lower your gaze from her, because there is no obedience to a creature in disobedience to the Creator.