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JEE Main & Advanced Mathematics Vector Algebra Question Bank Scalar triple product and their applications

  • question_answer
    If the given vectors \[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] \[({{a}^{2}}+ac,\,-ac,\,{{c}^{2}}+ac)\] and \[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] are coplanar, where none of a, b and c is zero, then

    A)             \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\]

    B)             \[bc+ca+ab=0\]

    C)             \[a+b+c=0\]

    D)             \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\]

    Correct Answer: B

    Solution :

                    Accordingly, \[\left| \,\begin{matrix}    -bc & {{b}^{2}}+bc & {{c}^{2}}+bc  \\    {{a}^{2}}+ac & -ac & {{c}^{2}}+ac  \\    {{a}^{2}}+ab & {{b}^{2}}+ab & -ab  \\ \end{matrix}\, \right|=0\]                 \[\Rightarrow {{(ab+bc+ca)}^{3}}=0\Rightarrow ab+bc+ca=0.\]


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