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Question
If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of \[\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} .\]
Solution
Given: D, E, F are the midpoints of the sides BC, CA, AB respectively.
Then, the position vectors of the midpoints D, E, F are given by \[\frac{\overrightarrow{b} + \overrightarrow{c}}{2}, \frac{\overrightarrow{c} + \overrightarrow{a}}{2}, \frac{\overrightarrow{a} + \overrightarrow{b}}{2}\]
\[\text{ Now, }\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \left( \frac{\overrightarrow{b} + \overrightarrow{c}}{2} \right) - \overrightarrow{a} + \left( \frac{\overrightarrow{c} + \overrightarrow{a}}{2} \right) - \overrightarrow{b} + \left( \frac{\overrightarrow{a} + \overrightarrow{b}}{2} \right) - \overrightarrow{c} \]
\[ = 2\left( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{2} \right) - \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right)\]
\[ = \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right) - \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right)\]
\[ = \overrightarrow{0}\]
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