Depends on what you want to do.
You need to have done Vector Calculus and Differential Equations to do the new Applied Maths major beginning 2014. They're replacing the third year subject, Partial Differential Equations, with a second year subject called Differential Equations. Vector Calculus is a prereq for Diff Eq, and Diff Eq itself is a prereq for two of the new subjects they're introducing for the Applied Maths major.
So yeah, you'd have to do both of those subjects if you wanted to go into Applied Maths starting from 2014. Unfortunately the handbook hasn't made much of a mention of it at all as of yet.
I don't know too much about Engineering Mathematics, just that it's a prereq for some of the engineering system majors. Not sure if Vector Calculus is as well, though.