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.

Answer 1

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,

\[\overrightarrow{AB} + \overrightarrow{AC} = \lambda \overrightarrow{AD} \]

\[ \Rightarrow \overrightarrow{b} - \overrightarrow{a} + \overrightarrow{c} - \overrightarrow{a} = \lambda \left( \frac{\overrightarrow{b} + \overrightarrow{c}}{2} - \overrightarrow{a} \right)\]

\[ \Rightarrow \overrightarrow{b} + \overrightarrow{c} - 2 \overrightarrow{a} = \lambda \left( \frac{\overrightarrow{b} + \overrightarrow{c} - 2 \overrightarrow{a}}{2} \right)\]

\[ \Rightarrow \lambda = 2\]