Bound on $|x \cdot y|$ from below

Question

Take $x$ and $y$ to be real $n$-dimensional vectors with an angle $\theta$ between them. Also take "$\cdot$" to mean the real dot product. We can bound $|x \cdot y|$ from above pretty easily by doing: \begin{align} |x\cdot y| &= |x||y||\cos \theta| \\ &\leq |x||y| \end{align} since $|\cos \theta | \leq 1$. You can bound $|x \cdot y|$ from below by $0$ since $|\cos \theta| \geq 0$. It would appear $0$ is as good as it gets using this method but does anyone have an expression which bounds $|x \cdot y|$ from below using $|x|$ and $|y|$?

Answer 1

The answer is no - if you take any orthogonal basis vectors $b_1$ and $b_2$, then $b_1 \cdot b_2 = 0$, so if $x$ and $y$ are scalar multiples of $b_1$ and $b_2$ respectively then $|x \cdot y| = |x||y||b_1 \cdot b_2| = 0$.