Correlation : How is correlation analogous to dot product?
So using your formula for the cross-correlation:

$$R_{xy}(n)=\sum_m x(m)y(n+m)$$  

we see that 

$$\begin{align}
R_{xy}[n]&amp;=\sum_m x[m]y[n+m]\\
&amp;= \sum_m x[m] \tilde y_n[m]\\
&amp;= x \cdot \tilde y_n\\
\text{for}\\
\tilde y_n[k] &amp;= y[n+k]\text{ ,}\\
\end{align}$$ 
which is a dot product.

So for any "shift" in the cross-correlation, you have the dot product of the first operand with the $n$-shifted second operand.

Note, by the way, that your formulas are only correct for real-valued $x$, i.e. in general, the correct formula for dot product and cross correlation are:

$$\begin{align}
x\cdot y &amp;= \sum\limits_{i=0}^{N-1} x^*[i]y[i]\\
R_{xy}[n] &amp;= \sum\limits_{i=0}^{N-1} x^*[i]y[n+i]\\
\text{with}\\
x,y &amp;\in \mathbb C^N\\
{z}^*&amp;= \Re\{z\} - \Im\{z\} 
\end{align}$$