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. (1) only (B) is correct (2) neither (A) nor (B) is correct (3) only (A) is correct (4) both (A) and (B) are correct.
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.\\
(1) only (B) is correct\\
(2) neither (A) nor (B) is correct\\
(3) only (A) is correct\\
(4) both (A) and (B) are correct.