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JEE Main & Advanced Mathematics Vector Algebra Question Bank Scaler or Dot product of two vectors and its application

  • question_answer
    If the position vectors of the points A, B, C, D be \[\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,2\,\mathbf{i}+5\,\mathbf{j},\,\,3\,\mathbf{i}+2\,\mathbf{j}-3\mathbf{k}\]and \[\mathbf{i}-6\,\mathbf{j}-\mathbf{k},\] then the angle between the vectors \[\overrightarrow{AB}\] and \[\overrightarrow{CD}\] is

    A)             \[\frac{\pi }{4}\]

    B)             \[\frac{\pi }{3}\]

    C)             \[\frac{\pi }{2}\]

    D)             \[\pi \]

    Correct Answer: D

    Solution :

               \[\overrightarrow{AB}=\mathbf{i}+4\mathbf{j}-\mathbf{k},\] \[\overrightarrow{CD}=-2\mathbf{i}-8\mathbf{j}+2\mathbf{k}\]                    \[\cos \theta =\frac{\overrightarrow{AB}.\overrightarrow{CD}}{|\overrightarrow{AB}|.|\overrightarrow{CD}|}=\frac{-2-32-2}{\sqrt{18}.\sqrt{72}}\]                             \[=\frac{-2-32-2}{2\times 18}=-1\Rightarrow \theta =\pi .\]


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