Two lists were created using only letters
listV = {{a, b, c}, {d, e, f}, {g, h, i}}
$\left(
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \\
\end{array}
\right)$
listU = {{j, k, l}, {m, n, o}, {p, q, r}}
$\left(
\begin{array}{ccc}
j & k & l \\
m & n & o \\
p & q & r \\
\end{array}
\right)$
With this command it is possible to create all the possibilities of sum of vectors (Modified)
allVectors = Flatten[Outer[Plus, listV, listU], 2]
$\left(
\begin{array}{ccc}
a+j & a+k & a+l \\
a+m & a+n & a+o \\
a+p & a+q & a+r \\
b+j & b+k & b+l \\
b+m & b+n & b+o \\
b+p & b+q & b+r \\
c+j & c+k & c+l \\
c+m & c+n & c+o \\
c+p & c+q & c+r \\
d+j & d+k & d+l \\
d+m & d+n & d+o \\
d+p & d+q & d+r \\
e+j & e+k & e+l \\
e+m & e+n & e+o \\
e+p & e+q & e+r \\
f+j & f+k & f+l \\
f+m & f+n & f+o \\
f+p & f+q & f+r \\
g+j & g+k & g+l \\
g+m & g+n & g+o \\
g+p & g+q & g+r \\
h+j & h+k & h+l \\
h+m & h+n & h+o \\
h+p & h+q & h+r \\
i+j & i+k & i+l \\
i+m & i+n & i+o \\
i+p & i+q & i+r \\
\end{array}
\right)$
Now the same previous information applying values to the vectors
numericListV = {{1, 2, 0}, {0, 2, 4}, {5, 8, 10}}
$\left(
\begin{array}{ccc}
1 & 2 & 0 \\
0 & 2 & 4 \\
5 & 8 & 10 \\
\end{array}
\right)$
Graphics3D[{Red,
MapThread[Arrow, {{{0, 0, 0}, #}} & /@ numericListV // Transpose]}]

numericListU = {{1, 4, 5}, {3, 5, 7}, {8, 8, 9}}
$\left(
\begin{array}{ccc}
1 & 4 & 5 \\
3 & 5 & 7 \\
8 & 8 & 9 \\
\end{array}
\right)$
Graphics3D[{Blue,
MapThread[Arrow, {{{0, 0, 0}, #}} & /@ numericListU // Transpose]}]

allVectorsNumeric =
Flatten[Outer[Plus, numericListV, numericListU], 2]
$\left(
\begin{array}{ccc}
2 & 5 & 6 \\
4 & 6 & 8 \\
9 & 9 & 10 \\
3 & 6 & 7 \\
5 & 7 & 9 \\
10 & 10 & 11 \\
1 & 4 & 5 \\
3 & 5 & 7 \\
8 & 8 & 9 \\
1 & 4 & 5 \\
3 & 5 & 7 \\
8 & 8 & 9 \\
3 & 6 & 7 \\
5 & 7 & 9 \\
10 & 10 & 11 \\
5 & 8 & 9 \\
7 & 9 & 11 \\
12 & 12 & 13 \\
6 & 9 & 10 \\
8 & 10 & 12 \\
13 & 13 & 14 \\
9 & 12 & 13 \\
11 & 13 & 15 \\
16 & 16 & 17 \\
11 & 14 & 15 \\
13 & 15 & 17 \\
18 & 18 & 19 \\
\end{array}
\right)$
Graphics3D[{Green,
MapThread[
Arrow, {{{0, 0, 0}, #}} & /@ allVectorsNumeric // Transpose]}]

Flatten[Outer[Plus, ListV, ListU], 1]
$\endgroup$V2+U1, ... V2+U1, V2+U2
. Do you meanV2+U1, ... V3+U1, V3+U2
? $\endgroup$