(a) False. (b) False.
Work Step by Step
(a) False. One can define infinitely many inner products in $R^n$, for example $$\langle u,v\rangle =t u\cdot v, \quad u,v \in R^n, \quad t\in R.$$ (b) False. The condition $$\langle u,u\rangle=0 \Longleftrightarrow u=0$$ shows that the only vector which has zero norm is the zero vector.