The answer to the question above would be that a word or phrase that modifies a word not included in the sentence is a dangling modifier.
A modifier changes, alters, limits, or adds more information as regards something else in the sentence. Modifiers are considered dangling when it is not clear what is being modified. The word or phrase modified is not clearly stated in the sentence. In fact, dangling modifers occur when the word or words they should modify have been omitted from the sentence. Take A:
a) Hoping to excuse my lateness, the note was written and given to my boss.
"Hoping to excuse my lateness" is a dangling modifier. The note cannot hope anything because it is an inanimate object, the words that "hoping to excuse my lateness" modify have been omitted.
Correction: Hoping to excuse my lateness, I wrote a note and gave it to my boss.