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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The vector \[\mathbf{a}+\mathbf{b}\] bisects the angle between the vectors a and b, if

    A) \[|\mathbf{a}|\,=\,|\mathbf{b}|\]

    B) \[|\mathbf{a}|\,=\,|\mathbf{b}|\] or angle between a and b is zero

    C) \[|\mathbf{a}|\,\,=m\,|\mathbf{b}|\]

    D) None of these

    Correct Answer: B

    Solution :

    • Since the angle between \[\mathbf{a}+\mathbf{b}\] and \[\mathbf{a}\]and the angle between \[\mathbf{a}+\mathbf{b}\] and \[\mathbf{b}\]are the same, so we have           
    • \[\frac{(\mathbf{a}+\mathbf{b})\,.\,\mathbf{a}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}=\frac{(\mathbf{a}+\mathbf{b})\,.\,\mathbf{b}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}\]                     
    • \[\Rightarrow \frac{|\mathbf{a}{{|}^{2}}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}+\frac{\mathbf{b}\,.\,\mathbf{a}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{a}|}=\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}+\frac{|\mathbf{b}{{|}^{2}}}{|\mathbf{a}+\mathbf{b}|\,|\mathbf{b}|}\]                    \[\Rightarrow \frac{|\mathbf{a}|-|\mathbf{b}|}{|\mathbf{a}+\mathbf{b}|}\left( 1-\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|} \right)=0\]                   
    • Hence \[|\mathbf{a}|\,=\,|\mathbf{b}|\] or angle between \[\mathbf{a}\] and \[\mathbf{b}\] is 0.


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