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

  • question_answer
    Let a, b, c be distinct non-negative numbers. If the vectors \[a\mathbf{i}+a\mathbf{j}+c\mathbf{k},\,\,\mathbf{i}+\mathbf{k}\] and \[c\mathbf{i}+c\mathbf{j}+b\mathbf{k}\] lie in a plane, then c is [IIT 1993; AIEEE 2005]

    A)             The arithmetic mean of a and b

    B)           The geometric mean of a and b

    C)             The harmonic mean of a and b

    D)             Equal to zero

    Correct Answer: B

    Solution :

                    \[\,\left| \,\begin{matrix}    a & a & c  \\    1 & 0 & 1  \\    c & c & b  \\ \end{matrix}\, \right|=0\Rightarrow \left| \,\begin{matrix}    a & 0 & c  \\    1 & -1 & 1  \\    c & 0 & b  \\ \end{matrix}\, \right|=0\] {Applying \[{{C}_{2}}\to {{C}_{2}}-{{C}_{1}}\}\]                                 \[\Rightarrow a\,(-b)+c\,(c)=0\Rightarrow {{c}^{2}}=ab.\]                                 Hence \[c\] is the geometric mean of \[a\] and \[b.\]


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