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

  • question_answer
    The value of b such that scalar product of the vectors \[(\mathbf{i}+\mathbf{j}+\mathbf{k})\] with the unit vector parallel to the sum of the vectors \[(2\mathbf{i}+4\mathbf{j}-5\mathbf{k})\] and \[(b\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is 1, is [MNR 1992; Roorkee 1985, 95; Kurukshetra CEE 1998; UPSEAT 2000]

    A)             ? 2

    B)             ? 1

    C)             0

    D)             1

    Correct Answer: D

    Solution :

               Parallel vector \[=(2+b)\mathbf{i}+6\mathbf{j}-2\mathbf{k}\]            Unit vector \[=\frac{(2+b)\mathbf{i}+6\mathbf{j}-2\mathbf{k}}{\sqrt{{{b}^{2}}+4b+44}}\]            According to the condition, \[1=\frac{(2+b)+6-2}{\sqrt{{{b}^{2}}+4b+44}}\]                 \[\Rightarrow {{b}^{2}}+4b+44={{b}^{2}}+12b+36\]\[\Rightarrow 8b=8\Rightarrow b=1.\]


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