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

  • question_answer
    If \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\]where a, b, c are three non-coplanar vectors, then the value of \[(\mathbf{a}+\mathbf{b}+\mathbf{c})\,.\,(\mathbf{p}+\mathbf{q}+\mathbf{r})\] is given by   [MNR 1992; UPSEAT 2000]

    A)             3

    B)             2

    C)             1

    D)             0

    Correct Answer: A

    Solution :

                    \[\mathbf{p}+\mathbf{q}+\mathbf{r}=\frac{\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}+\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\]                 \[(\mathbf{a}+\mathbf{b}+\mathbf{c})\,.\,(\mathbf{p}+\mathbf{q}+\mathbf{r})=\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]+[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]+[\mathbf{c}\,\mathbf{a}\,\mathbf{b}]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}=3\].


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