Final answer:
When a woman asks you out on a date, it typically means she is interested in getting to know you better and possibly pursuing a romantic relationship. Such an invitation can be a casual yet significant step toward developing a closer connection.
Step-by-step explanation:
When a woman suggests you go on a date or asks you out, it generally means that she is interested in spending time with you in a social setting to get to know you better. It represents an opportunity to build a personal connection and possibly explore the potential for a romantic relationship. The phrase "Well, why don't you come to dinner to-morrow night?" implies an informal invitation which could be a casual step towards forming a closer bond. The response to such an invitation can indeed set a new direction in one's life, signifying the beginning of a significant personal or emotional journey.