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)

only $( A )$ is correct

Answer 2

∣a+b+c∣2=∣a+b−c∣2
2a⋅b+2b⋅c+2c⋅a=2a⋅b−2b⋅c−2c⋅a
4a⋅c=0
B is incorrect
∣a+λc∣2≥∣a∣2
λ2c2≥0
True ∀λ∈R (A) is correct.