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, \[\overrightarrow{AB}\,.\,\overrightarrow{CD}+\overrightarrow{BC}\,.\,\overrightarrow{AD}+\overrightarrow{CA}\,.\,\overrightarrow{BD}\] \[=\mathbf{a}\,.\,\mathbf{c}+\mathbf{b}\,.(\mathbf{a}+\mathbf{b}+\mathbf{c})+(-\mathbf{a}-\mathbf{b})\,.\,(\mathbf{b}+\mathbf{c})\] \[=\mathbf{a}\,.\,\mathbf{c}+\mathbf{b}\,.\,\mathbf{a}+\mathbf{b}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}-\mathbf{a}\,.\,\mathbf{b}-\mathbf{a}\,.\,\mathbf{c}-\mathbf{b}\,.\,\mathbf{b}-\mathbf{b}\,.\,\mathbf{c}\]\[=0\].You need to login to perform this action.
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