How does Geometric Duality Theorem work?

I got to the end of Geometric Duality Theorem which is

$$\vec c = \sum y_i^*\vec \alpha+\sum w_j^*(-\vec e_j)$$

$$\vec c$$ = Original objective function being maximized.
$$y_i^*$$ = Optimal solution to dual problem
$$w_j^*$$ = Optimal slack variable solutions to primal problem
$$\vec e_j$$ = A unit vector where the jth index is 1

I don't really understand what the negative on $$e_j$$ means. How does that help get the original objective function when multiplied by $$w_j^*$$? Also when you sum them do they just become a vector? So you just do a vector sum(for example, [1,2,3][1,2,1]+[1,2,3][1,2,1]=[1,4,3]+[1,4,3]=[2,8,6]?)