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This line comes from "The Things They Carried" by Tim O Brien when he served in Vietnam. The term hard vocabulary to contain terrible softness simply reflects what he has done to serve in Vietnam. In times of war, it is not good to be internally soft as it may serve as weakness to enemies. Therefore, hard vocabulary refers to the words that one would say to himself and others in order to project strength and hide softness or weakness inside.
A problem I have with someone is arguing and we do it all the time but I'm trying to stop. I can take care easily my stress it goes away quickly.
No they aren't proper to use in writing class they are considered slang. It would show your teacher that your not interested in sounding educated and proper.
Hope this helps!