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 \[|\vec{a}+\vec{b}+\vec{c}|.\]

Answer:

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)] \[\Rightarrow \]      \[{{(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})}^{2}}={{(3)}^{2}}+{{(4)}^{2}}+{{(5)}^{2}}\] \[\Rightarrow \]   \[|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}=50\] \[\Rightarrow \]   \[|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}=\sqrt{50}=5\sqrt{2}\]