Question

Let $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ be the vectors of length 3, 4, 5 respectively. Let $\overrightarrow{a}$ be perpendicular to $\overrightarrow{b}$ + $\overrightarrow{c}$, $\overrightarrow{b}$ to $\overrightarrow{c}$ + $\overrightarrow{a}$ and $\overrightarrow{c}$ to $\overrightarrow{a}$ + $\overrightarrow{b}$. Then |$\overrightarrow{a}$ + $\overrightarrow{b}$ + $\overrightarrow{c}$| is equal to :

• A. $2\sqrt{5}$
• B. $2\sqrt{2}$
• C. $10\sqrt{5}$
• D. $5\sqrt{2}$

Answer 1

The correct option is D $5\sqrt{2}$

$\left|\overline{a}\right|=3\left|\overline{b}\right|=4\left|\overline{c}\right|=5$

angle between $\overline{a}$ and $\overline{b}$ be $\alpha$

angle between $\overline{b}$ and $\overline{c}$ be $\beta$

angle between $\overline{c}$ and $\overline{a}$ be $\gamma$

$\therefore \overline{a}.(\overline{b}+\overline{c})=0\overline{a}.\overline{b}+\overline{a}.\overline{c}=0\left|\overline{a}\right|\left|\overline{b}\right|\cos \alpha +\left|\overline{a}\right|\left|\overline{c}\right|\cos \gamma =0\therefore \cos \alpha =0\cos \gamma =0\overline{b}.(\overline{c}+\overline{a})=0\overline{b}.\overline{c}+\overline{b}.\overline{a}=0\left|\overline{b}\right|\left|\overline{c}\right|\cos \beta +\left|\overline{b}\right|\left|\overline{a}\right|\cos \alpha =0$

$\therefore \cos \alpha =0\cos \beta =0\cos \alpha =\cos \beta =\cos \gamma =0$

$\therefore \overline{a},\overline{b},\overline{c}$ are all perpendicular to each other.

$|\overline{a}+\overline{b}+\overline{c}|=\sqrt{3^2+4^2+5^2}=\sqrt{50}=5\sqrt{2}$