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

  • question_answer
    \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a};\,\,\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b};\,\,\mathbf{a}\ne 0;\,\,\mathbf{b}\ne 0;\,\,\mathbf{a}\ne \lambda \mathbf{b},\,\,\]a  is not perpendicular to b, then \[\mathbf{r}=\]                                     [EAMCET 1993]

    A)                 \[\mathbf{a}-\mathbf{b}\]

    B)                 \[\mathbf{a}+\mathbf{b}\]

    C)                 \[\mathbf{a}\times \mathbf{b}+\mathbf{a}\]

    D)                 \[\mathbf{a}\times \mathbf{b}+\mathbf{b}\]

    Correct Answer: B

    Solution :

               \[\mathbf{r}\times \mathbf{a}-\mathbf{b}\times \mathbf{a}=0\] and \[\mathbf{r}\times \mathbf{b}+\mathbf{b}\times \mathbf{a}=0\]            Adding, we get \[\mathbf{r}\times (\mathbf{a}+\mathbf{b})=0\]                 But as we are given \[\mathbf{a}\ne \lambda \,\mathbf{b},\] therefore \[\mathbf{r}=\mathbf{a}+\mathbf{b}.\]


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