Orthogonal families are independent

Orthogonality. For $i\neq j$, $\langle v_i,v_j\rangle=0$.

Linear independence. Suppose $\sum_{i=1}^k\alpha_i v_i=0$. For a fixed $j$:
$$0=\langle 0,v_j\rangle=\left\langle\sum_i\alpha_i v_i,v_j\right\rangle=\sum_i\alpha_i\langle v_i,v_j\rangle=\alpha_j\|v_j\|^2.$$
Since $v_j\neq0$, $\|v_j\|^2>0$, so $\alpha_j=0$.

Conclusion. All coefficients are zero, so the family is linearly independent.