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JEE Main & Advanced Mathematics Vector Algebra Question Bank Vector or Cross product of two vectors and its application

  • question_answer
    If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{a}\,.\,\mathbf{c},\,\,\mathbf{a}\,\times \mathbf{b}=\mathbf{a}\times \mathbf{c}\] and \[\mathbf{a}\ne \mathbf{0},\] then     [RPET 1990]

    A)                 \[\mathbf{b}=\mathbf{0}\]

    B)                 \[\mathbf{b}\ne \mathbf{c}\]

    C)                 \[\mathbf{b}=\mathbf{c}\]

    D)                 None of these

    Correct Answer: C

    Solution :

                    \[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=0,\] \[\mathbf{a}\times (\mathbf{b}-\mathbf{c})=\mathbf{0}\Rightarrow \mathbf{b}=\mathbf{c}\] or \[\mathbf{a}=\mathbf{0},\] but \[\mathbf{a}\ne \mathbf{0}.\] Hence \[\mathbf{b}-\mathbf{c}=\mathbf{0}.\] i.e., \[\mathbf{b}=\mathbf{c}\].


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