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He was vying for his approval.
I was vying with two people i know from school.
Here is an example using the word "vying":
"Peter and Daniel were both vying for the affections of their classmate, Annette."
Vying means competing. It's an adjective describing a person who likes to compete and is willing to fight.
Example:
Many swimmers were vying for the title
Competition is vying with others.