Question

If $a=3,b=4,c=5$ and each one of $\overrightarrow{a},\overrightarrow{b}$ & $\overrightarrow{c}$ is perpendicular to the sum of the remaining, then $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$ is equal to

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

Answer 1

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

According to the question,
$\overrightarrow{a}\cdot (\overrightarrow{b}+\overrightarrow{c})=0\cdots \left(1\right)$
$\overrightarrow{b}\cdot (\overrightarrow{a}+\overrightarrow{c})=0\cdots \left(2\right)$
$\overrightarrow{c}\cdot (\overrightarrow{a}+\overrightarrow{b})=0\cdots \left(3\right)$
Adding $\left(1\right),\left(2\right)\text{and}\left(3\right),$ we get
$2(\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a})=0\cdots \left(4\right)$

$\text{Now},|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{|}^2=(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$
$=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+|\overrightarrow{c}{|}^2\left[\text{From}\right(4\left)\right]$
$=9+16+25=50$
$\therefore |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=\sqrt{50}=5\sqrt{2}$