Let a , b , c be three vectors such that a (b+c), b (c+a) and c (… - Vidyadip

MCQ

Let $\overline{ a }, \overline{ b }, \overline{ c }$ be three vectors such that $\bar{a} \perp(\bar{b}+\bar{c}), \bar{b} \perp(\bar{c}+\bar{a})$ and $\bar{c} \perp(\bar{a}+\bar{b})$.
If $|\overline{ a }|=1,|\overline{b}|=2,|\overline{ c }|=3$, then $|\overline{ a }+\overline{ b }+\overline{ c }|$ is

A
$\sqrt{6}$
B
14
✓
$\sqrt{14}$
D
6

✓

Answer

Correct option: C.

$\sqrt{14}$

(C) Given, $\bar{a} \perp(\bar{b}+\bar{c}), \bar{b} \perp(\bar{c}+\bar{a})$ and $\bar{c} \perp(\bar{a}+\bar{b})$
$\begin{array}{l}\Rightarrow \overline{ a } \cdot \overline{ b }+\overline{ a } \cdot \overline{ c }=0, \overline{b} \cdot \overline{ c }+\overline{ b } \cdot \overline{ a }=0, \overline{ c } \cdot \overline{ a }+\overline{ c } \cdot \overline{ b }=0 \\ \Rightarrow \overline{ a } \cdot \overline{ b }+\overline{ b } \cdot \overline{ c }+\overline{ c } \cdot \overline{ a }=0\end{array}$
Now, $|\overline{ a }+\overline{ b }+\overline{ c }|^2=|\overline{ a }|^2+|\overline{ b }|^2+|\overline{ c }|^2$ $+2(\overline{ a } \cdot \overline{ b }+\overline{ b } \cdot \overline{ c }+\overline{ c } \cdot \overline{ a })$
$\begin{array}{l}\Rightarrow|\overline{ a }+\overline{ b }+\overline{ c }|^2=1+4+9=14 \\ \Rightarrow|\overline{ a }+\overline{ b }+\overline{ c }|=\sqrt{14}\end{array}$

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