Pensieve header: Mathematica is ridiculous when you add vectors not as it likes. Adding $n$ zero vectors seems to take an exponential amount of time!

t/: a_t+b_t:=t[a[[1]]+b[[1]]]

Table[First@Timing@Sum[t[0],{n}], {n,18}]

r/: r[a_]+r[b_]:=r[a+b]

Total[Table[r[0],{200}]]

z/:a_z+b_z:=z@@Table[a[[k]]+b[[k]],{k,Length[a]}]

z[0,0,0]+z[0,0,0]

Table[First@Timing@Sum[z[0,0,0],{n}], {n,18}]

y/:Plus[ys___y]:=Module[{l=Min[Length/@{ys}]},
y@@Total[Take[List@@#,l]&/@{ys}]]

y[0,0,0]+y[0,0]

y[0,0,0]+y[0,0,0]

Table[First@Timing@Sum[y[0,0,0],{n}], {n,200}]

Timing@Sum[y[0,0,0],{10000}]