Cauchy-Schwarz inequality

Quadratic. For all $t\in\mathbb{R}$,
$$Q(t)=\sum_{i=1}^n(a_it+b_i)^2=\left(\sum a_i^2\right)t^2+2\left(\sum a_ib_i\right)t+\left(\sum b_i^2\right)\ge0.$$

Discriminant. A quadratic with positive leading coefficient that is $\ge0$ everywhere has discriminant $\le0$:
$$4\left(\sum a_ib_i\right)^2-4\left(\sum a_i^2\right)\left(\sum b_i^2\right)\le0.$$
Hence the inequality.

Equality case. Equality iff the quadratic has a double root, i.e. $a_it+b_i=0$ for all $i$ for some $t$, i.e. the vectors $(a_i)$ and $(b_i)$ are proportional.