Day Questions

Is it permissible to brush my teeth with water and toothpaste if I am fasting?

It is permissible, provided the water and/or toothpaste does not pass the throat.

If someone makes a nadhr (promise) to Allah to write a book, may he put the material into a video or audio presentation instead, or does he have to write a book?

This depends on the intention of the person that made the vow (nadhr). If he meant particularly and literally writing a book, then that is specifically what he must do. If he meant bringing some material or information into the public or to publish something in any way, then he has a choice.