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

  • question_answer
    A, B, C, D are any four points, then \[\overrightarrow{AB}\,\,.\,\,\overrightarrow{CD}\,\,+\,\overrightarrow{\,BC}\,\,.\,\,\overrightarrow{AD}\,\,+\overrightarrow{CA}\,\,.\,\,\overrightarrow{BD}\,\,=\]            [MNR 1986]

    A)             \[2\,\,\overrightarrow{AB}\,\,.\,\,\overrightarrow{BC}\,\,.\,\,\overrightarrow{CD}\]

    B)             \[\overrightarrow{AB}\,\,+\,\,\overrightarrow{BC}\,\,+\,\,\overrightarrow{CD}\]

    C)             \[5\sqrt{3}\]

    D)             0

    Correct Answer: D

    Solution :

    \[\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\mathbf{a}+\mathbf{b}+\mathbf{c}\]                    \[\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\mathbf{a}+\mathbf{b}\] or \[\overrightarrow{CA}=-(\mathbf{a}+\mathbf{b})\]                    \[\overrightarrow{BD}=\overrightarrow{BC}+\overrightarrow{CD}=\mathbf{b}+\mathbf{c}\]                    Therefore, \[\overrightarrow{AB}\,.\,\overrightarrow{CD}+\overrightarrow{BC}\,.\,\overrightarrow{AD}+\overrightarrow{CA}\,.\,\overrightarrow{BD}\]                 \[=\mathbf{a}\,.\,\mathbf{c}+\mathbf{b}\,.(\mathbf{a}+\mathbf{b}+\mathbf{c})+(-\mathbf{a}-\mathbf{b})\,.\,(\mathbf{b}+\mathbf{c})\]                      \[=\mathbf{a}\,.\,\mathbf{c}+\mathbf{b}\,.\,\mathbf{a}+\mathbf{b}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}-\mathbf{a}\,.\,\mathbf{b}-\mathbf{a}\,.\,\mathbf{c}-\mathbf{b}\,.\,\mathbf{b}-\mathbf{b}\,.\,\mathbf{c}\]\[=0\].


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