Vector Algebra

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) \(|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|\) for all \(\lambda \in \mathbb{R}\).

(B) \(\vec{a}\) and \(\vec{c}\) are always parallel.

Then,