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JEE Main & Advanced Mathematics Vector Algebra Question Bank Modulus of vector Algebra of vectors

  • question_answer
    Let A and B be points with position vectors a and b with respect to the origin O. If the point C on OA is such that \[2AC=CO,\,\,CD\] is parallel to OB and \[|\overrightarrow{CD}|\,\,=\,\,3|\overrightarrow{OB}|,\] then \[\overrightarrow{AD}\] is equal to

    A) \[3\mathbf{b}-\frac{\mathbf{a}}{2}\]               

    B) \[3\mathbf{b}+\frac{\mathbf{a}}{2}\]

    C) \[3\mathbf{b}-\frac{\mathbf{a}}{3}\]

    D) \[3\mathbf{b}+\frac{\mathbf{a}}{3}\]

    Correct Answer: C

    Solution :

    Since \[\overrightarrow{OA}=\mathbf{a},\] \[\overrightarrow{OB}=\mathbf{b}\] and \[2AC=CO\] By section formula \[\overrightarrow{OC}=\frac{2}{3}\mathbf{a}.\]         Therefore, \[|\overrightarrow{CD}|=3|\overrightarrow{OB}|\,\Rightarrow \overrightarrow{CD}=3\mathbf{b}\] \[\Rightarrow \overrightarrow{OD}=\overrightarrow{OC}+\overrightarrow{CD}=\frac{2}{3}\mathbf{a}+3\mathbf{b}\] Hence, \[\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=\frac{2}{3}\mathbf{a}+3\mathbf{b}-\mathbf{a}=3\mathbf{b}-\frac{1}{3}\mathbf{a}.\]          


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