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STD 12 - 11. three dimension geometry — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 11. three dimension geometry4 Marks
Question
If three mutually perpendicular lines have direction cosines $({l_1},{m_1},{n_1}),({l_2},{m_2},{n_2})$ and $({l_3},{m_3},{n_3})$, then the line having direction cosines ${l_1} + {l_2} + {l_3}$, ${m_1} + \,\,{m_2} + \,\,{m_3}$ and ${n_1} + {n_2} + {n_3}$ make an angle of ..…… $^o$ with each other 
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Answer

a
(a) Lines are mutually perpendicular

 ${l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2} = 0,\,\,\,{l_2}{l_3} + {m_2}{m_3} + {n_2}{n_3} = 0$

and ${l_1}{l_3} + {m_1}{m_3} + {n_1}{n_3} = 0$

Therefore, $\,\theta = {\cos ^{ - 1}}[({l_1} + {l_2} + {l_3})\,{l_1} + ({m_1} + {m_2} + {m_3})\,{m_1}$

$ + ({n_1} + {n_2} + {n_3}){n_1}]$

==> $\theta = {\cos ^{ - 1}}\,\left[ {\sum {l_1^2} } \right] = {\cos ^{ - 1}}\,(1)\,\, $

$\Rightarrow \,\,\theta = {0^o}$

Similarly with other lines, it will make ${0^o}$ angle.

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