Vector~!!!

(a) OD = OA/3 = a/3.
CD = OD - OC = a/3 - c.
CF = mCD = m(a/3 - c) ( m = lambda)
so OF = OC + CF = c + m(a/3 - c) = ma/3 + (1 - m)c.............(1)
(b) BA = -OC = - c.
BE = 2BA/5 = - 2c/5.
OE = OC + CB + BE = c + a - 2c/5 = a + 3c/5.
(c) Let OF = nOE.
so OF = na + 3nc/5............(2)
(1) = (2) we get
n = m/3 and
3n/5 = 1 - m.
From these 2 equations we get m = lambda = 5/6 and n = 5/18.
(d)
Area of triangle OCF : area of triangle OFD = CF : FD = 5 : 1.
Area of triangle OCF : area of triangle CFE = OF : FE = 5 : 13.
so area of triangle OFD : area of triangle CFE = 1 : 13.