Question

In a triangle OAC, if B is the mid-point of side AC and \[\overrightarrow{OA} = \overrightarrow{a} , \overrightarrow{OB} = \overrightarrow{b}\], then what is \[\overrightarrow{OC}\].

Answer 1

In ∆OAC, \[\overrightarrow{OA} = \overrightarrow{a}\] and \[\overrightarrow{OB} = \overrightarrow{b}\]
It is given that B is the mid-point of AC.
∴ Position vector of B = \[\frac{\text{ Position vector of A + Position vector of C }}{2}\] 

\[\Rightarrow \overrightarrow{OB} = \frac{\overrightarrow{OA} + \overrightarrow{OC}}{2}\]

\[ \Rightarrow \overrightarrow{b} = \frac{\overrightarrow{a} + \overrightarrow{OC}}{2}\]

\[ \Rightarrow \overrightarrow{a} + \overrightarrow{OC} = 2 \overrightarrow{b} \]

\[ \Rightarrow \overrightarrow{OC} = 2 \overrightarrow{b} - \overrightarrow{a}\]