Question

If $\overline{a},\overline{b},\overline{c}$ are three vectors such that $\left|\overline{a}\right|=5,\left|\overline{b}\right|=12,\left|\overline{c}\right|=13$ and $\overline{a},\overline{b},\overline{c}$ are perpendicular to $\overline{b}+\overline{c},\overline{c}+\overline{a},\overline{a}+\overline{b}$ respectively, then $|\overline{a}+\overline{b}+\overline{c}|=$

• A. $\frac{1}{\sqrt{13}}$
• B. $2\sqrt{13}$
• C. $13\sqrt{2}$
• D. $\frac{\sqrt{2}}{13}$

Answer 1

Answer: (C)

$13\sqrt{2}$

According to the condition given that $\overline{a}\perp \overline{b}+\overline{c}$ , $\overline{b}\perp \overline{a}+\overline{c},\overline{c}\perp \overline{a}+\overline{b}$ ; we get that $\overline{a}.\overline{b}=\overline{b}.\overline{c}=\overline{a}.\overline{c}=0$
Thus, $|\overline{a}+\overline{b}+\overline{c}|=\sqrt{a^2+b^2+c^2+2(\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a})}=\sqrt{a^2+b^2+c^2}=\sqrt{338}=13\sqrt{2}$