Question

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

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

Answer 1

Answer: (D)

$5\sqrt{2}$

According to question,

$\overrightarrow{a}.(\overrightarrow{b}+\overrightarrow{c})=0$

$\overrightarrow{b}.(\overrightarrow{c}+\overrightarrow{a})=0$

$\overrightarrow{c}.(\overrightarrow{a}+\overrightarrow{b})=0$

$\Rightarrow \overrightarrow{a}.\overrightarrow{b}+\overrightarrow{a}.\overrightarrow{c}=0\to \left(1\right)$

$\Rightarrow \overrightarrow{b}.\overrightarrow{c}+\overrightarrow{b}.\overrightarrow{a}=0\to \left(2\right)$

$\Rightarrow \overrightarrow{c}.\overrightarrow{a}+\overrightarrow{c}.\overrightarrow{b}=0\to \left(3\right)$

${|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|}^2={\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2+2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{a}.\overrightarrow{c})\to \left(4\right)$

Adding (1), (2) and (3),

$2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{a}.\overrightarrow{c})=0(\because \overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{a})$

$\Rightarrow {|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|}^2={\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2+0$

$\Rightarrow {|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|}^2=9+16+25=50$

$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=5\sqrt{2}$