Yes, there is a connection between correlation and dot products (also called inner products).

Consider the vector space of real-valued random variables. These random variables don't have to be independent, so they may have a covariance

[math]X\cdot Y=\mbox{Cov}(X,Y)=E((X-\mu_X)(Y-\mu_Y)).\tag*{}[/math]

Covariance is bilinear, so it can be used to define an inner product on the vector space of random variables making it an inner product space.

With an inner product, you can define the norm of a vector (also called the length of the vector by as the square root of the inner product with itself. So [math]\|X\|=\sqrt{X\cdot X}[/math] is defined as the square root of [math]\mbox{Cov}(X,X)=E((X-\mu_X)^2).[/math] This norm is the standard deviation [math]\sigma_X[/math] of [math]X[/math].

The correlation of two random variables is defined by

[math]\rho_{XY}=\dfrac{\mbox{Cov}(X,Y)}{\sigma_X\sigma_Y}\tag*{}[/math]

That is precisely the definition of the the cosine of the angle [math]\theta[/math] between two vectors

[math]\cos\theta=\dfrac{X\cdot Y}{\|X\|\,\|Y\|}.\tag*{}[/math]

In summary, covariance is an inner product, standard deviations are norms, and correlations are cosines of angles.