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JEE Main & Advanced Mathematics Vector Algebra Question Bank Self Evaluation Test - Vector Algebra

  • question_answer
    If a, b, c are the \[{{p}^{th}},\text{ }{{q}^{th}}.\text{ }{{\text{r}}^{th}}\] terms of an HP and \[\vec{u}=(q-r)\vec{i}+(r-p)\vec{j}+(p-q)\vec{k},\vec{v}=\frac{{\vec{i}}}{a}+\frac{{\vec{j}}}{b}+\frac{{\vec{k}}}{c}\] then

    A)  \[\vec{u},\vec{v}\] are parallel vectors

    B) \[\vec{u},\vec{v}\] are orthogonal vectors

    C) \[\vec{u}.\vec{v}=1\]

    D) \[\vec{u}\times \vec{v}=\vec{i}+\vec{j}+\vec{k}\]

    Correct Answer: B

    Solution :

    [b] \[\frac{1}{a}=A+(p-1)D;\frac{1}{b}=A+(q-1)D;\] \[\frac{1}{c}=A+(r-1)D\] \[\therefore q-r=\frac{c-b}{bcD},r-p=\frac{a-c}{acD}\] \[p-q=\frac{b-a}{abD}\Rightarrow \frac{q-r}{a}+\frac{r-p}{b}+\frac{p-q}{c}=0\] \[\Rightarrow \overset{\to }{\mathop{u}}\,\cdot \overset{\to }{\mathop{v}}\,=0\]


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