Q.The inequality |a·b| ≤ |a||b| is called

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality bounds the dot product of two vectors by the product of their magnitudes. For any two vectors $\vec a$ and $\vec b$,

Equality holds exactly when the vectors are parallel (one is a scalar multiple of the other). The inequality follows immediately from the geometric formula $\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta$, since $|\cos\theta| \le 1$; but it is a genuinely deeper fact because it also holds in the component form $\left(\sum a_i b_i\right)^2 \le \left(\sum a_i^2\right)\left(\sum b_i^2\right)$ for real numbers, independently of geometry. …