How to repesent n x m multiplication into symbol notation?

Question

I am not a mathematician and so I might not be using the right terms. I have a vector of n components and another vector of m components

x = (x1, x2, ... xn)
y = (y1, y2, ... ym)

How do I represent the following sum of products using sigma and/or pi notations

SOP = x1y1 + x1y2 +... x1ym+....+ xny1 + xny2 + ... + xnym

In other words the sum of all possible products of pairs between x and y.

Answer 1

One way would be $$ \sum_{i=1}^{n}\sum_{j=1}^{m}x_{i}y_{j} $$

Answer 2

You have two choices:

$$\displaystyle\sum_{\substack{1\leq i\leq n \\ 1\leq j\leq m}} x_iy_j$$ or

$$ \sum_{i=1}^{n}\sum_{j=1}^{m}x_{i}y_{j} $$