• # question_answer If $\vec{a},\,\,\vec{b}$ and $\vec{c}$ are three vectors, such that $|\vec{a}|\,\,=3,$ $|\vec{b}|\,\,=4$ and $|\vec{c}|\,\,=5$ and each one of these is perpendicular to the sum of other two, find $|\vec{a}+\vec{b}+\vec{c}|.$

Given, $|\vec{a}|\,\,=3,$$|\overrightarrow{b}|\,\,=4,$$|\vec{c}|\,\,=5$ According to the question each one of given three vectors perpendicular to the sum of other two. $\therefore$      $\overrightarrow{a}\bot (\overrightarrow{b}+),$$\overrightarrow{b}\bot (\overrightarrow{c}+),$$\overrightarrow{c}\bot (\overrightarrow{a}+)$ $\Rightarrow$   $\overrightarrow{a}\cdot (\overrightarrow{b}+)=0$                                      ?(i) $\overrightarrow{b}\cdot (\overrightarrow{c}+)=0$                                     ?(ii) and       $\overrightarrow{c}\cdot (\overrightarrow{a}+)=0$                                   ?(iii) On adding Eqs. (i), (ii) and (iii), we get $2\,(\overrightarrow{a}\cdot +\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a})=0$                        ?(iv) Now, ${{(\overrightarrow{a}++\overrightarrow{c})}^{2}}={{(\overrightarrow{a})}^{2}}+{{(\overrightarrow{b})}^{2}}+{{(\overrightarrow{c})}^{2}}$ $+2\,(\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a})$ $\Rightarrow$${{(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})}^{2}}=\,\,|\overrightarrow{a}{{|}^{2}}+|\overrightarrow{b}{{|}^{2}}+|\overrightarrow{c}{{|}^{2}}+0$ [using Eq. (iv)] $\Rightarrow$      ${{(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})}^{2}}={{(3)}^{2}}+{{(4)}^{2}}+{{(5)}^{2}}$ $\Rightarrow$   $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}=50$ $\Rightarrow$   $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}=\sqrt{50}=5\sqrt{2}$