Question

Let O be the centre of regular hexagon ABCDEF. Find the sum.
$OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space plus space stack OE space with rightwards arrow on top space plus space OF with rightwards arrow on top$ .

Solution

We have
   $OA with rightwards arrow on top space equals space minus OD with rightwards arrow on top. space space OB with rightwards arrow on top space equals space minus OE with rightwards arrow on top stack OC space with rightwards arrow on top space equals space minus OF with rightwards arrow on top$
$therefore space space space space OA with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 0 with rightwards arrow on top$ ...(1)
   $OB with rightwards arrow on top space plus space OE with rightwards arrow on top space equals space 0 with rightwards arrow on top$ ...(2)
   $OC with rightwards arrow on top space plus space OF with rightwards arrow on top space equals 0 with rightwards arrow on top$ ...(3)
Adding (1), (2), (3), we get,
   $OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus OC with rightwards arrow on top space plus space OD with rightwards arrow on top space plus space OE with rightwards arrow on top space plus space OF with rightwards arrow on top space equals space 0 with rightwards arrow on top$