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Let  \vec{a}=2\hat{i}+\hat{j}+\hat{k},  and  \hat{b}  and \hat{c} be two nonzero vectors such that   \left|\hat{a}+\hat{b}+\hat{c}\right|= \left|\hat{a}+\hat{b}+\hat{c}\right|   and

 \hat{b}\, .\, \hat{c}=0  Consider the following two statements:

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

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

 

Option: 1

 both (A) and (B) are correct 


Option: 2

 only (A) is correct
 


Option: 3

neither (A) nor (B) is correct 


Option: 4

only (B) is correct


Answers (1)

best_answer

\begin{aligned} & |\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|, \vec{b} \cdot \vec{c}=0 \\ & |\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}+\vec{b}-\vec{c}|^2 \\ & |\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{a} \cdot \vec{c} \\ & =|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2 \vec{a} \cdot \vec{b}-2 \vec{b} \cdot \vec{c}-2 \vec{a} \cdot \vec{c} \\ & 2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{a} \cdot \vec{c}=2 \vec{a} \cdot \vec{b}-2 \vec{b} \cdot \vec{c}-2 \vec{a} \cdot \vec{c} \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}} \\ \end{aligned}\begin{aligned} & \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}}=0 \text { (B is incorrect) } \\ & \end{aligned}

$$ \begin{aligned} & |\vec{a}+\lambda \vec{c}|^2 \geq|\vec{a}|^2 \\ & |\vec{a}|^2+\lambda^2|\vec{c}|^2+2 \lambda \vec{a} \cdot \vec{c} \geq|\vec{a}|^2 \\ & =\lambda^2 c^2 \geq 0 \end{aligned}

True  \forall \lambda \in \mathrm{R} \quad  (A is correct) 

 

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