Let $ \langle \cdot, \cdot \rangle: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} $ be an inner product on the vector space $ \mathbb{R}^n $ over $ \mathbb{R} $. Consider the following statements:
$P:$ $ |\langle \mathbf{u}, \mathbf{v} \rangle| \leq \frac{1}{2} (\langle \mathbf{u}, \mathbf{u} \rangle + \langle \mathbf{v}, \mathbf{v} \rangle) $ for all $ \mathbf{u}, \mathbf{v} \in \mathbb{R}^n $.
$Q:$If $ \langle \mathbf{u}, \mathbf{v} \rangle = \langle 2\mathbf{u}, -\mathbf{v} \rangle $ for all $ \mathbf{v} \in \mathbb{R}^n $, then $ \mathbf{u} = \mathbf{0} $.
Then the correct option is:
(A) both P and Q are TRUE
(B) P is TRUE and Q is FALSE
(C) P is FALSE and Q is TRUE
(D) both P and Q are FALSE
