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

  • question_answer
    Let \[\overset{\to }{\mathop{a}}\,,\overset{\to }{\mathop{b}}\,,\overset{\to }{\mathop{c}}\,\] be non-coplanar vectors and\[\overset{\to }{\mathop{p}}\,=\frac{\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]},\,\,\,\overset{\to }{\mathop{q}}\,=\frac{\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]},\,\,\,\overset{\to }{\mathop{r}}\,=\frac{\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}.\] What is the value of\[(\vec{a}-\vec{b}-\vec{c}).\vec{p}+(\vec{b}-\vec{c}-\vec{a}).\vec{q}+(\vec{c}-\vec{a}-\vec{b}).\vec{r}\]?

    A) 0

    B) -3

    C) 3

    D) -9

    Correct Answer: C

    Solution :

    [c] As given \[\overset{\to }{\mathop{p}}\,=\frac{\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]},\overset{\to }{\mathop{q}}\,=\frac{\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]},\] and
    \[\overset{\to }{\mathop{r}}\,=\frac{\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}\]
    \[\therefore (\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{c}}\,).\overset{\to }{\mathop{p}}\,+(\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{c}}\,-\overset{\to }{\mathop{a}}\,).\overset{\to }{\mathop{q}}\,+(\overset{\to }{\mathop{c}}\,-\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{b}}\,).\overset{\to }{\mathop{r}}\,\]
    \[=\frac{a.(\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,)}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}+\frac{\overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,)}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}+\frac{\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,)}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}\]
    [Since \[\overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,)=0,\]
    \[\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,),=0,\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,)=0\overset{\to }{\mathop{a}}\,.(\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,)]\]
    \[=0,\overset{\to }{\mathop{a}}\,.(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,)=0\] and \[\overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,)=0\]
    \[=\frac{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}+\frac{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{c}{\mathop{c}}\,]}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}+\frac{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}{[\overset{\to }{\mathop{a}}\,\overset{\to }{\mathop{b}}\,\overset{\to }{\mathop{c}}\,]}=3\]


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