Question

In a triangle OAC, if B is the mid-point of side AC and $\overrightarrow{\mathrm{OA}}=\overrightarrow{a},\overrightarrow{\mathrm{OB}}=\overrightarrow{b}$, then what is $\overrightarrow{\mathrm{OC}}$? [CBSE 2015]

Answer 1

In ∆OAC, $\overrightarrow{\mathrm{OA}}=\overrightarrow{a}$ and $\overrightarrow{\mathrm{OB}}=\overrightarrow{b}$.



It is given that B is the mid-point of AC.

∴ Position vector of B = $\frac{\mathrm{Position}\mathrm{vector}\mathrm{of}\mathrm{A}+\mathrm{Position}\mathrm{vector}\mathrm{of}\mathrm{C}}{2}$

$$\begin{gathered} \Rightarrow \overrightarrow{\mathrm{OB}}=\frac{\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}}{2} \\ \Rightarrow \overrightarrow{\mathit{b}}=\frac{\overrightarrow{\mathit{a}}+\overrightarrow{\mathrm{OC}}}{2} \\ \Rightarrow \overrightarrow{\mathit{a}}+\overrightarrow{\mathrm{OC}}=2\overrightarrow{b} \\ \Rightarrow \overrightarrow{\mathrm{OC}}=2\overrightarrow{b}-\overrightarrow{a} \end{gathered}$$