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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2005

  • question_answer
        If the vectors\[\overrightarrow{a},\overrightarrow{b}\]and\[\overrightarrow{c}\]from the sides BC, CA and AB respectively, of a triangle ABC then:

    A)  \[\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{b}=0\]

    B)  \[\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}\]

    C)  \[\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0\]

    D)  \[\overrightarrow{a}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=0\]

    Correct Answer: B

    Solution :

                                    We have\[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0\] \[\overrightarrow{a}+\overrightarrow{b}=-\overrightarrow{c}\] \[\Rightarrow \]               \[\overrightarrow{a}+\overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{c}=0\] \[\Rightarrow \]               \[\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}\] Similarly \[\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}\] Hence  \[\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}\]


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