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

  • question_answer
    Let \[\vec{a},\vec{b}\] and \[\vec{c}\] be three non-zero vectors such that no two of these are collinear. If the vector \[\vec{a}+2\vec{b}\] is collinear with \[\vec{c}\] and \[\vec{b}+3\vec{c}\] is collinear with \[\vec{a}\] (\[\lambda \] being some non-zero scalar) then \[\vec{a}+2\vec{b}+6\vec{c}\] equals

    A) 0

    B) \[\lambda \vec{b}\]

    C) \[\lambda \vec{c}\]

    D) \[\lambda \vec{a}\]

    Correct Answer: C

    Solution :

    [c] Let \[\vec{a}+2\vec{b}=t\vec{c}\] and \[\vec{b}+3\vec{c}=s\vec{a}\], where t and s are scalars. Adding, we get \[\vec{a}+3\vec{b}+3\vec{c}=t\vec{c}+s\vec{a}\Rightarrow \vec{a}+2\vec{b}+6\vec{c}\] \[=t\vec{c}+s\vec{a}-\vec{b}+3\vec{c}\] \[=t\vec{c}+(\vec{b}+3\vec{c})-\vec{b}+3\vec{c}=(t+6)\vec{c}\] [using \[s\,\,\vec{a}=\vec{b}+3\vec{c}\]] \[=\lambda \vec{c}\],     where \[\lambda =t+6\]


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