Question

OABCDE is a regular hexagon. The points A and B have position vectors `bar"a"` and `bar"b"` respectively referred to the origin O. Find, in terms of `bar"a"` and `bar"b"` the position vectors of C, D and E.

Answer 1

Given: `bar"OA" = bar"a", bar"OB" = bar"b"`

Let AD, BE, OC meet at M.

Then M bisects AD, BE, OC.

`bar"AB" = bar"AO" + bar"OB"`

`= - bar"OA" + bar"OB"`

`= - bar"a" + bar"b" = bar"b" - bar"a"`

∵ OABM is a parallelogram

∴ `bar"OM" = bar"AB" = bar"b" - bar"a"`

`bar"OC" = 2bar"OM" = 2(bar"b" - bar"a") = 2bar"b" - 2bar"a"`

`bar"OD" = bar"OC" + bar"CD"`

`= bar"OC" - bar"DC"`

`= bar"OC" - bar"OA"`    ...[∵ OA = DC and OA || DC]

`= 2bar"b" - 2bar"a" - bar"a"`

`= 2bar"b" - 3bar"a"`

`bar"OE" = bar"OM" + bar"ME"`

`= (bar"b" - bar"a") - bar"EM"`

`= bar"b" - bar"a" - bar"a"`   ....[∵ EM = OA and EM || OA]

`= bar"b" - 2bar"a"`

Hence, the position vectors of C, D and E are `2bar"b" - 2bar"a", 2bar"b" - 3bar"a"  "and"  bar"b" - 2bar"a"` respectively.