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Question
If D is the mid-point of side BC of a triangle ABC such that \[\overrightarrow{AB} + \overrightarrow{AC} = \lambda \overrightarrow{AD} ,\] write the value of λ.
Solution
Given: D is the midpoint of the side BC of a triangle ABC such that \[\overrightarrow{AB} + \overrightarrow{BC} = \lambda \overrightarrow{AD} .\]
Let \[\overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c}\] are the position vectors of AB, BC and CA.
Now, the position vector of D is \[\frac{\overrightarrow{b} + \overrightarrow{c}}{2}\]. Then,
\[\overrightarrow{AB} = \overrightarrow{b} - \overrightarrow{a} \]
\[ \overrightarrow{AC} = \overrightarrow{c} - \overrightarrow{a} \]
\[ \overrightarrow{AD} = \frac{\overrightarrow{b} + \overrightarrow{c}}{2} - \overrightarrow{a}\]
Now, we have,
\[\overrightarrow{AB} + \overrightarrow{AC} = \lambda \overrightarrow{AD} \]
\[ \Rightarrow \overrightarrow{b} - \overrightarrow{a} + \overrightarrow{c} - \overrightarrow{a} = \lambda \left( \frac{\overrightarrow{b} + \overrightarrow{c}}{2} - \overrightarrow{a} \right)\]
\[ \Rightarrow \overrightarrow{b} + \overrightarrow{c} - 2 \overrightarrow{a} = \lambda \left( \frac{\overrightarrow{b} + \overrightarrow{c} - 2 \overrightarrow{a}}{2} \right)\]
\[ \Rightarrow \lambda = 2\]
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