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JEE Main & Advanced Mathematics Vector Algebra Question Bank Mock Test - Vector Algebra

  • question_answer
    Let \[\vec{a}=\hat{i}+\hat{j};\hat{b}=2\hat{i}-\hat{k}\]. Then vector \[\vec{r}\] satisfying the equations \[\vec{r}\times \vec{a}=\vec{b}\times \vec{a}\]and \[\vec{r}\times \vec{b}=\vec{a}\times \vec{b}\] is

    A) \[\hat{i}-\hat{j}+\hat{k}\]         

    B) \[3\hat{i}-\hat{j}+\hat{k}\]

    C) \[3\hat{i}+\hat{j}-\hat{k}\]        

    D) \[\hat{i}-\hat{j}-\hat{k}\]

    Correct Answer: C

    Solution :

    [c] \[\vec{r}\times \vec{a}=\vec{b}\times \vec{a}\] or \[(\vec{r}-\vec{b})\times \vec{a}=0\] \[\vec{r}\times \vec{b}=\vec{a}\times \vec{b}\] or \[(\vec{r}-\vec{b})\times \vec{b}=0\] If \[\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\], then \[\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    x-2 & y & z+1  \\    1 & 1 & 0  \\ \end{matrix} \right|=0\] and \[\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    x-1 & y-1 & z  \\    2 & 0 & -1  \\ \end{matrix} \right|=0\] \[\Rightarrow z+1=0,x-y=2\] And \[y-1=0,x-1+2z=0\] \[\Rightarrow x=3,y=1,z=-1\]


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