Let a =2 i + j + k, and b and c be two nonzero vectors such that | a + b + c |=| a + b - c | and b · c =0. Consider the following two statements: (A)…

Let $\displaystyle \vec{a}=2 \hat{i}+\hat{j}+\hat{k}$, and $\displaystyle \vec{b}$ and $\displaystyle \vec{c}$ be two nonzero vectors such that $\displaystyle |\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\displaystyle \vec{b} \cdot \vec{c}=0$. Consider the following two statements:

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

$\displaystyle \vec{a}$ and $\displaystyle \vec{c}$ are always parallel.

Then,