Dot product and linear regression

I'm studying PCA and my professor said something about finding the linear regression by doing the dot product of both axis. Could someone explain to me why? The dot product returns a number. What's the relationship between that number and the linear regression?

In my example, I have two vectors

$stat\_grade = [0,1,3,7,10]$

$physics\_grade = [1,5,8,10,10]$

The first step is normalizing them:

$ \frac{stat\_grade - mean(stat\_grade)}{std(stat\_grade)} = [-1.69131435 -0.52489066 0.34992711 0.93313895 0.93313895]$

$ \frac{physics\_grade - mean(physics\_grade)}{std(physics\_grade)} = [-1.11613741 -0.85039041 -0.3188964 0.7440916 1.54133261]$

Computing the dot product between the normalized vectors will return $4.355129045906131$

I can't understand how a single number can help me to find the linear regression

Thanks for the response!