Question

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 $\lambda \in R$.
(B) $\vec{a}$ and $\vec{c}$ are always parallel. Then. is

• A. both (A) and (B) are correct
• B. only $( A )$ is correct
• C. only (B) is correct
• D. neither $( A )$ nor $(B)$ is correct

Answer 1

Answer: (B)

$ |\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}+\vec{b}-\vec{c}|^2$
$ 2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{c} \cdot \vec{a}=2 \vec{a} \cdot \vec{b}-2 \vec{b} \cdot \vec{c}-2 \vec{c} \cdot \vec{a}$
$4 \vec{a} \cdot \vec{c}=0$
$B$ is incorrect
$ |\vec{ a }+\lambda \vec{ c }|^2 \geq|\vec{ a }|^2 $
$ \lambda^2 c ^2 \geq 0$
True $\forall \lambda \in R$ (A) is correct.