Convince me Gabriel's Horn is possible
Jelly, 9 8 bytes
ṙ1×æ.Ɱ⁵Ḥ

Try it online!
Takes input as [z, y, x] as allowed by OP.
-1 byte thanks to Nick Kennedy
We calculate
$$2\left( \begin{matrix} z\times y \\ y\times x \\ x\times z\end{matrix} \right) \cdot \left( \begin{matrix} i \\ 1 \\ 1 \end{matrix} \right) = 2(yzi + yx+xz)$$
for each \$1 \le i \le 10\$
How it works
ṙ1×æ.Ɱ⁵Ḥ - Main link. Takes [z, y, x] on the left
ṙ1       - Rotate once left; [y, x, z]
  ×      - Product; [yz, xy, xz]
      ⁵  - Yield 10
     Ɱ   - For each integer 1 ≤ i ≤ 10:
   æ.    -   Dot product. Pad to [i, 1, 1] and calculate
              [yz, xy, xz] . [i, 1, 1] = yzi+xy+xz
       Ḥ - Double each