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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    If a, b and c are unit vectors, then \[|\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}}\]  does not exceed [IIT Screening 2001]

    A) 4

    B) 9

    C) 8

    D) 6

    Correct Answer: B

    Solution :

    • \[|\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}}\]                   
    • \[=2({{\mathbf{a}}^{2}}+{{\mathbf{b}}^{2}}+{{\mathbf{c}}^{2}})-2(\mathbf{a}.\mathbf{b}+\mathbf{b}\,.\mathbf{c}+\mathbf{c}\,.\,\mathbf{a})\]                   
    • \[=2\times 3-2(\mathbf{a}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}+\mathbf{c}\,.\,\mathbf{a})\]                   
    • \[=6-\{{{(\mathbf{a}+\mathbf{b}+\mathbf{c})}^{2}}-{{\mathbf{a}}^{2}}-{{\mathbf{b}}^{2}}-{{\mathbf{c}}^{2}}\}\]\[=9-|\mathbf{a}+\mathbf{b}+\mathbf{c}{{|}^{2}}\le 9.\]


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