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

  • question_answer
    \[\overset{\to }{\mathop{a}}\,\text{ },\overset{\to }{\mathop{b}}\,\,\,and\,\,\vec{c}\] are three vectors with magnitude \[|\overset{\to }{\mathop{a}}\,|=4,|\overset{\to }{\mathop{b}}\,|=4,|\overset{\to }{\mathop{c}}\,|=2\] and such that \[\overset{\to }{\mathop{a}}\,\] is perpendicular to  is perpendicular to \[(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,),\overset{\to }{\mathop{b}}\,\] is  perpendicular to \[(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,)\] and \[\overset{\to }{\mathop{c}}\,\] is perpendicular to \[(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,)\]. It follows that \[|\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,|\] is equal to:

    A) 9

    B) 6

    C) 5

    D) 4

    Correct Answer: B

    Solution :

    [b] Since, \[\overset{\to }{\mathop{a}}\,,\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{c}}\,\] are three vectors with magnitude \[|\overset{\to }{\mathop{a}}\,|=|\overset{\to }{\mathop{b}}\,|=4\] and \[|\overset{\to }{\mathop{c}}\,|=2,\]
    As \[\overset{\to }{\mathop{a}}\,\] is perpendicular to \[(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)\]
    \[\Rightarrow \overset{\to }{\mathop{a}}\,.(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)=0\] or \[\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{c}}\,)=0\]     ?(i)
    \[\overset{\to }{\mathop{b}}\,\] is perpendicular to \[(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,)\]
    \[\Rightarrow \overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,)=0\] or \[\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{a}}\,=0\]                 ?(ii)
    \[\Rightarrow \overset{\to }{\mathop{c}}\,\] is perpendicular to \[(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,)\]
    \[\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,)=0\] or \[\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{b}}\,=0\]                     ?(iii)
    From equations (i), (ii) and (iii), we get
    \[\Rightarrow 2(\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{a}}\,)=0\]
    Further we know that
    \[|\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,{{|}^{2}}=|\overset{\to }{\mathop{a}}\,{{|}^{2}}+|\overset{\to }{\mathop{b}}\,{{|}^{2}}+|\overset{\to }{\mathop{c}}\,{{|}^{2}}\]
    \[+\overrightarrow{2a}.\vec{b}+\overrightarrow{2b}.\overrightarrow{c}+\overrightarrow{2c}.\overrightarrow{a}\]
    \[\Rightarrow |\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,{{|}^{2}}={{4}^{2}}+{{4}^{2}}+{{2}^{2}}+0=36\]
    or \[|\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,|=6\]


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