## The three musketeers

Given vectors $\mathbf{x}$ and $\mathbf{y}$, the three quantities $\mathbf{x}\cdot\mathbf{x}$, $\mathbf{x}\cdot\mathbf{y}$ and $\mathbf{y}\cdot\mathbf{y}$ are very interesting indeed.

In fact, the concept of inner product vector space revolves largely around those quantities. The underlying mathematical fact is the Cauchy-Schwarz inequality $(\mathbf{x}\cdot\mathbf{y})^2 \le (\mathbf{x}\cdot\mathbf{x})\times (\mathbf{y}\cdot\mathbf{y})$, which is a corollary of the axioms of an inner product vector space.