JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    If \[\alpha \,(\mathbf{a}\times \mathbf{b})+\beta \,(\mathbf{b}\times \mathbf{c})+\gamma \,(\mathbf{c}\times \mathbf{a})=\mathbf{0}\] and at least one of the numbers \[\alpha ,\,\,\beta \] and \[\gamma \] is non-zero, then the vectors a, b and c are                                  

    A) Perpendicular

    B) Parallel

    C) Coplanar

    D) None of these

    Correct Answer: C

    Solution :

    • Let \[\alpha \ne 0,\] then \[\alpha (\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}+\beta (\mathbf{b}\times \mathbf{c})\,.\,\mathbf{c}+\gamma (\mathbf{c}\times \mathbf{a})\,.\,\mathbf{c}=0\,\]                   
    • \[\Rightarrow \alpha [\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=0\Rightarrow [\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=0\], \[\left\{ \because \,\alpha \ne 0 \right\}\]   
    • Hence a, b, c are coplanar.

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