Let $\vec{a}, \vec{b}, \vec{c}$ be vectors of lengths $3,4,5$ respectively. Let $\vec{a}$ be perpendicular to $\vec{b}+\vec{c}, \vec{b}$ be perpendicular to $\vec{c}+\vec{a}$ and $\vec{c}$ be perpendicular to $\vec{a}+\vec{b}$, then the length of vector $\vec{a}+\vec{b}+\vec{c}$ is

• A

  $5$
• B

  $5 \sqrt{3}$
• Correct answer
  C

  $5 \sqrt{2}$
• D

  $5 \sqrt{6}$

Solution

$\begin{aligned} & \text { Now }|\vec{a}+\vec{b}+\vec{c}|=\sqrt{|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{c} \cdot \vec{a}} \\ & =\sqrt{3^2+4^2+5^2+0} \quad[\because|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5 \text { given and from(iv) } \\ & =\sqrt{50} \\ & =5 \sqrt{2}\end{aligned}$

Hence, the correct answer is option: 'C'.