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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 \[\mathbf{a}=2\mathbf{i}+3\mathbf{j}-5\mathbf{k},\,\,\mathbf{b}=m\mathbf{i}+n\mathbf{j}+12\mathbf{k}\] and \[\mathbf{a}\times \mathbf{b}=0,\] then \[(m,\,\,n)=\]    

    A)                 \[\left( -\frac{24}{5},\,\frac{36}{5} \right)\]       

    B)                 \[\left( \frac{24}{5},\,-\frac{36}{5} \right)\]

    C)                 \[\left( -\frac{24}{5},\,-\frac{36}{5} \right)\]     

    D)                 \[\left( \frac{24}{5},\,\frac{36}{5} \right)\]

    Correct Answer: C

    Solution :

               \[\mathbf{a}\times \mathbf{b}=\left| \begin{matrix}    \mathbf{i} & \mathbf{j} & \mathbf{k}  \\    2 & 3 & -5  \\    m & n & 12  \\ \end{matrix} \right|\]                          \[=(36+5n)\,\mathbf{i}-(24+5m)\,\mathbf{j}+(2n-3m)\,\mathbf{k}=0\]                                 \[\Rightarrow m=\frac{-24}{5},\,\,\,n=\frac{-36}{5}\].


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