$$ABCDEF$$   is a regular hexagon where centre $$O$$ is the origin. If the position vectors of $$A$$ and $$B$$ are $$\overrightarrow i - \overrightarrow j + 2\overrightarrow k $$   and $$2\overrightarrow i + \overrightarrow j + \overrightarrow k $$   respectively then $$\overrightarrow {BC} $$  is equal to :

Solution :

$$\eqalign{ & \overrightarrow {OA} = \overrightarrow i - \overrightarrow j + 2\overrightarrow k \,\,\,\,\,\,\,\,\,\,\overrightarrow {OB} = 2\overrightarrow i + \overrightarrow j + \overrightarrow k \cr & \therefore \overrightarrow {OC} = \overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} = \overrightarrow i + 2\overrightarrow j - \overrightarrow k \cr & \therefore \,\overrightarrow {BC} = \overrightarrow {OC} - \overrightarrow {OB} = \left( {\overrightarrow i + 2\overrightarrow j - \overrightarrow k } \right) - \left( {2\overrightarrow i + \overrightarrow j + \overrightarrow k } \right) \cr & = - \overrightarrow i + \overrightarrow j - 2\overrightarrow k \cr} $$