dot product

$$\bf{x}\dot\bf{y}=|\bf{x}||\bf{y}|cos(\theta)$$

For center y and standardized X, find the w that maximize the sample correlation between y and Xw, linear combination of columns of X.

$$correlation = \frac{y\hat{y}}{ \|y\| \|\hat{y}\|}, where\ \hat{y}=Xw$$

This is the cosine between y and a vector in the column space of X, the projection of y to column space of X is the vector maximize it. Thus, $\hat{\beta}^{OLS}$ would be the optimized w.