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That's a very strange diagram. The equalities you wrote down are definitely true for the case where all three vectors are in the same plane as you said, but just to make sure you understand, these forumulas are also true for any three vectors.
eh... if anyone is interested in the solution, here it is:
Yes I know how to prove it algebraically... just not with the diagram.
There are two ways to explain this. Either geometrically or algebraically.
See the image, show using the diagram...