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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 20

  • question_answer
    If  \[\overrightarrow{a},\,\,\overrightarrow{b},\,\,\overrightarrow{c}\] are three unit vectors such that \[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0},\] where \[\overrightarrow{0}\]  is null vector, then \[\overrightarrow{a}\,.\,\overrightarrow{b}+\overrightarrow{b}\,.\,\overrightarrow{c}+\overrightarrow{c}\,.\,\overrightarrow{a}\] is :

    A) \[-3\]               

    B)   \[-2\]      

    C)   \[-\frac{3}{2}\]                      

    D)   \[0\]

    Correct Answer: C

    Solution :

    We have \[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\] \[\therefore \,\,|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=0\Rightarrow |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}=0\] \[\Rightarrow \,\,|\overrightarrow{a}{{|}^{2}}+|\overrightarrow{b}{{|}^{2}}+|\overrightarrow{c}{{|}^{2}}+2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})=0\] \[\Rightarrow \,1+1+1+2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})=0\] \[\Rightarrow \,\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}=-\frac{3}{2}\]


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