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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Critical Thinking

  • question_answer
    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  ..... with each other

    A) \[0{}^\circ \]

    B) \[30{}^\circ \]

    C) \[60{}^\circ \]

    D) \[90{}^\circ \]

    Correct Answer: A

    Solution :

    • 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, \[\,\theta ={{\cos }^{-1}}[({{l}_{1}}+{{l}_{2}}+{{l}_{3}})\,{{l}_{1}}+({{m}_{1}}+{{m}_{2}}+{{m}_{3}})\,{{m}_{1}}\] \[+({{n}_{1}}+{{n}_{2}}+{{n}_{3}}){{n}_{1}}]\]           
    • Þ\[\theta ={{\cos }^{-1}}\,\left[ \sum{l_{1}^{2}} \right]={{\cos }^{-1}}\,(1)\,\,\Rightarrow \,\,\theta ={{0}^{o}}\]                   
    • Similarly with other lines, it will make \[{{0}^{o}}\] angle.


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