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Question
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the points A, B and C respectively, write the value of \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} .\]
Solution
Given: \[\overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c}\] are the position vectors of A, B, C respectively.
Then,
\[\overrightarrow{AB} = \overrightarrow{b} - \overrightarrow{a} \]
\[ \overrightarrow{BC} = \overrightarrow{c} - \overrightarrow{b} \]
\[ \overrightarrow{AC} = \vec{c} - \vec{a}\]
Therefore,
\[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} = \overrightarrow{b} - \overrightarrow{a} + \overrightarrow{c} - \overrightarrow{b} + \overrightarrow{c} - \overrightarrow{a} \]
\[ = 2 \left( \overrightarrow{c} - \overrightarrow{a} \right)\]
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