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JEE Main & Advanced Mathematics Vector Algebra Question Bank Mock Test - Vector Algebra

  • question_answer
    If \[\hat{a},\] \[\hat{b}\] and \[\hat{c}\] are three unit vectors, such that \[\hat{a}+\hat{b}+\hat{c}\] is also a unit vector and \[{{\theta }_{1}}\], \[{{\theta }_{2}}\] and \[{{\theta }_{3}}\] are angles between the vectors \[\hat{a}\], \[\hat{b}\]; \[\hat{b}\], \[\hat{c}\] and \[\hat{c}\], \[\hat{a}\], respectively, then among \[\theta _{1}^{{}}\],\[\theta _{2}^{{}}\]and \[\theta _{3}^{{}}\]

    A) all are acute angles

    B) all are right angles

    C) at least one is obtuse angle

    D) none of these

    Correct Answer: C

    Solution :

    [c] \[{{\left| \vec{a}+\vec{b}+\vec{c} \right|}^{2}}=1\] \[\Rightarrow {{\left| {\vec{a}} \right|}^{2}}+{{\left| {\vec{b}} \right|}^{2}}+{{\left| {\vec{c}} \right|}^{2}}+2\left| {\vec{a}} \right|\left| {\vec{b}} \right|\cos {{\theta }_{1}}\] \[+2\left| {\vec{b}} \right|\left| {\vec{c}} \right|\cos {{\theta }_{2}}+2\left| {\vec{c}} \right|\left| {\vec{a}} \right|\cos {{\theta }_{3}}=1\] \[\Rightarrow \cos {{\theta }_{1}}+\cos {{\theta }_{2}}+\cos {{\theta }_{3}}=-1\] Hence, one of \[{{\theta }_{1}},{{\theta }_{2}}\]and \[{{\theta }_{3}}\]should be an obtuse angle.


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