How Does Position Vector Calculation Work in Geometry?

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AD = AB + 2/3BC
OD - OA = OB - OA + 2/3(OC - OB)
OD = 1/3OB + 2/3OC

BE = BC + 3/4CA
BE = OC - OB + 3/4 OA - 3/4OC
OE = 1/4OC - OB + 3/4OA

Could someone please tell me if I am on the right track with this?
I am also stuck on the finding the position vector G. Whenever I attempt to find it, the G always cancels out leaving me with no equation left.
 

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Your doing ok so far. So if you represent each position vector as they suggested, e.g. OA=a, OB=b, OC=c, etc, you've got e=(3/4)a+(1/4)c and d=(1/3)b+(2/3)c, right? The line between A and D is given by a+t*(d-a) where t is a parameter which describes where you are on the line (e.g. if t=0 you are at a and if t=1 you are at d). Similarly the line between B and E is given by b+s*(e-d) for a different parameter s. You want to find the intersection of those two lines. I.e. find values of t and s that give the same point.