Let \( \vec{a} = 2\hat{i} + \hat{j} + \hat{k} \), and \( \vec{b} \) and \( \vec{c} \) be two nonzero vectors such that \( |\vec{a}+\vec{b}+\vec{c}| = |\vec{a}+\vec{b}-\vec{c}| \) and \( \vec{b}\cdot\vec{c} = 0 \). Consider the following two statements:
(A) \( |\vec{a}+\lambda \vec{c}| \ge |\vec{a}| \) for all \( \lambda \in \mathbb{R} \)
(B) \( \vec{a} \) and \( \vec{c} \) are always parallel.
