Solution.
By the rule of adding vectors we get
OA+AB=OB,
from here AB=OB−OA=b−a.
By the rule of adding vectors we get
OB+BC=OC,
from here BC=OC−OB=c−b.
D is the midpoint of BC, then BD=2c−b.
By the rule of adding vectors we get
OD=OB+BD=b+2c−b=2b+c.
By the rule of adding vectors we get
AD=AB+BD=b−a+2c−b=2b+c−2a.
E is the point on AD such that AE:ED=2:1, then AE=32AD=322b+c−2a=3b+c−2a.
By the rule of adding vectors we get
OE=OA+AE=a+3b+c−2a=3a+b+c.
Answer.
OD=2b+c
OE=3a+b+c
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