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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.
Solution
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.
Notes
[Note: Answer to `bar"OC"` in the textbook is incorrect.]
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